Mathematics 3 ESO - Teacher's book Unit 11 by Grupo Anaya, S.A.

Three-dimensional shapes

OVERVIEW OF THE UNIT

Three-dimensional shapes

POLYHEDRA

SOLIDS OF REVOLUTION

such as

Threedimensional shapes whose faces are polygons.

THE PRISM

the most important are REGULAR and SEMIREGULAR polyhedra

ALAT = p · h ATOT = ALAT + 2ABASE

V = ABASE · h

THE CUBOID

which are such as The more interesting are

whose surface is whose volume is THE PYRAMID

Three-dimensional shapes which are created by rotating a flat shape around an axis.

whose surface is whose volume is

ATOT = 2 · (ab + ac + bc)

V = ABASE · h = a · b · c

ALAT = p · a/2 ATOT = ALAT + ABASE V = 1 3 ABASE · h

ALAT = 2 π r h ATOT = 2 π r h + 2 π r2 V = ABASE · h = π r2 h

whose surface is whose volume is

THE CYLINDER

ALAT = π r g ATOT = π r g + π r2 whose surface is whose volume is

THE CONE V = 1 3 ABASE · h = 1 3 π r2 h

A = 4 π r2 whose surface is whose volume is V = 4 3 π r3

THE SPHERE

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Digital resources Inclusion and diversity Assessment

WHAT ARE WE GOING TO LEARN?

RESOURCES IN THE DIGITAL PROJECT

Initial pages Presentation: ‘What you need to know’.

Regular and semiregular polyhedra

• Regular polyhedra

• Duality in regular polyhedra

• Semiregular polyhedra

• Euler's formula

Truncating regular polyhedra

• Truncating at the midpoint of an edge

• Truncating leaving part of the edges

Planes of symmetry of a shape

• Planes of symmetry of a cube

• Planes of symmetry of prisms and cylinders

Axes of rotation of a shape

• Axes of rotation of a cube

• Axes of rotation of a tetrahedron

GeoGebra: Regular polyhedra.

GeoGebra: Truncating regular polyhedra.

GeoGebra: Draw three-dimensional shapes and trace their planes of symmetry. Planes of symmetry of a shape.

GeoGebra: Axes of rotation of a shape.

PEDAGOGICAL KEYS IN THE STUDENT’S BOOK

Technique: Intuition and deduction

Surface area of three-dimensional shapes

GeoGebra: Practise the area of the cylinder, the cone and the sphere. Practise the area of a truncated cone.

Technique: Questions bag Volume of three-dimensional shapes

GeoGebra: Practise calculating the areas and volumes of prisms, pyramids, cylinders and cones. Volume of a sphere.

Geographic coordinates

• Geographic coordinates

• Time zones

Final pages

• Exercises and problems solved

• Exercises and problems

• Maths workshop

• Self-assessment

Challenges that leave an imprint

• Think

• Test your skills

Video on the SDG commitment. Target 7.3. Answer key for the self-assessment. Documents to prepare a portfolio.

Final self-assessment. The essentials. Answer key for the exercises and problems. Assessment.

Personal dimension: Self-awareness

Skill: Speaking (descriptive text)

SDG Commitment. Target 7.3

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Unit presentation

This unit is dedicated to the study of three-dimensional shapes. Students will learn to analyse, describe and classify them, measure the lengths of their sides and calculate their surface areas and volumes.

Students are already familiar with the nomenclature of three-dimensional shapes will have already worked with their nets. They also already understand the concept of volume and the units of the metric system. However, all this content is still at the construction and learning stage, so it is not yet consolidated. Therefore, this unit should not be considered a review. Rather, it is a unit in which students consolidate and advance their knowledge of three-dimensional shapes.

We introduce the most common regular polyhedra and look in more detail at their duality. We describe how semiregular polyhedra can be formed by truncating regular polyhedra. We analyse symmetries. We discuss the indirect measurement of length and surface areas using what students have learnt about plane geometry, especially the Pythagorean theorem. We introduce some general procedures for calculating volumes. Finally, we study the terrestrial sphere, geographic coordinates and the consequences of the rotational movements of the Earth.

Basic knowledge

The basic knowledge we consider students should achieve is:

• The concept of a polyhedron. Nomenclature and classification.

• The concept of a solid of revolution. Nomenclature and classification.

• Using nomenclature for three-dimensional shapes to describe objects in the real world.

• Identifying and analysing regular and semiregular polyhedra.

• Identifying basic shapes and their most intuitive nets.

• Calculating the surface area and volume of some simple shapes from their nets or using a formula.

• Geographic coordinates. Parallels and meridians.

Complementary knowledge

In addition, it is useful for students to supplement their learning with other content, such as that listed below:

• Describing different three-dimensional shapes.

• Forming semiregular polyhedra through truncation.

• Identifying planes of symmetry and axes of rotation.

• Calculating lengths, surface areas and volumes in polyhedra and solids of revolution through reasoned procedures.

• Time zones.

Task preparation

• Resources to help students develop their spatial imagination and visualise the shapes they are studying.

• Objects that can be used to find areas and volumes: soft drink cans, bottles, glasses, prisms, etc.

• Blank spheres for hands-on activities, for example drawing lines of latitude and longitude.

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Three-dimensional shapes

STARTING THE UNIT

Students learn about some of the contributions made by Plato and Archimedes to culture and thought, as well as their respective approaches to science and mathematics.

For Archimedes, experimentation, technical methods and the construction of objects were a way of understanding physical phenomena or mathematical properties. This is in stark contrast to Plato’s disdain for practical experimentation.

After reading and listening to the text on page 226, students are encouraged to work on the speaking activity posed in the Language Bank box to improve their English speaking skills.

SOLVE APPLYING WHAT YOU ALREADY KNOW

Students are already familiar with polyhedra, bodies of revolution and the sections that can be made on them. This course will look at slightly more complex concepts, as well as reinforcing students’ previous knowledge.

In this section, in addition to reviewing some content, questions are proposed related to sections of three-dimensional shapes that can be answered with a reasonable spatial vision. Students can also use three-dimensional shapes made of plasticine. The investigation of sections of three-dimensional shapes can be very fruitful. Even in objects from their everyday life, students can practise finding interesting sections.

ANSWER KEY FOR ‘THINK’

1 Square: from the tetrahedron, cube, octahedron and dodecahedron.

Rectangle: from the tetrahedron, cube and dodecahedron.

Trapezoid: from the tetrahedron, cube, octahedron and dodecahedron.

In an octahedron, the polygon with the most sides that can be obtained is a hexagon, and the one with the fewest sides is a quadrilateral.

In a tetrahedron, the polygon with more sides is the quadrilateral, and the polygon with fewer sides is the triangle.

In a cube, the polygon with the most sides is the hexagon, and the polygon with the fewest sides is the triangle.

In a dodecahedron, the polygon with more sides is the decagon, and the polygon with fewer sides is the triangle.

In an icosahedron, the polygon with more sides is the decagon, and the polygon with fewer sides is the pentagon.

2 Pentagonal pyramid: B, C, F, I, Ñ

Cylinder: D, E, H, N, O

Half cylinder: A, D, E, E, H, L

Cone: B, E, G, I, K, M, N, O

3 With a small piece of cheese, you can make triangles, rectangles and squares.

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So, what role did these two geniuses (Plato and who was especially interested in moral philosophy, and considered science to be an inferior branch of abstractions and its separation from the material entrance of the Academy that he founded (a type of one ignorant of geometry enter’. in his honour) close relationship with the universe: thematics in its simplest form. Plato had great influence on subsequent thinking. mathematician and inventor. Throughout his life he designed and built multitude of ingenious discover physical and mathematical properties, obtain the areas and volumes of geometric shapes. He also studied the 13 solids that take his name. Although Plato would have probably been horrified of the greatest mathematicians of antiquity. Sove applying what you already know dimensional shape with plane. They’re flat shapes that can be polygonal or with curved sides. see some on the right. Remember what the regular polyhedra are. Let’s think about the sections we can create from them. For example, if we want octahedron and the icosahedron (strange, bearing mind that their faces are equilateral triangles). think And trapezium? Share your conclusions with your classmates. There are probably more than you think. sides that you can create with cross section? And with the least sides? Do the same with the other regular polyhedra. Let’s look at some other three-dimensional shapes. Do you know which of the sections on the right you can obtain from Use modelling clay to make three-dimensional shapes and create sections from them using piece of card or knife. If possible, try to obtain molds of the polyhedra and solids of Can you make triangles, squares and rectangles by cutting the cheese? Try with other objects you have at hand. Ñ THREE-DIMENSIONAL11SHAPES invented something? What was it? Tell partner. Language Bank

Developing thinking

and deduction’ technique for activity 2 on page 229. This technique is explained at anayaeducacion.es

Regular and semiregular polyhedra

SUGGESTED METHODOLOGY

While students are already familiar with regular polyhedra, there are some aspects we recommend working on:

• One interesting activity could be to find all the ways of constructing polyhedral angles using a regular polygon.

Students will find that an equilateral triangle can do this in three ways, a square, one, and a pentagon, another. This explains the existence of five, and only five, regular polyhedra.

• Another involves investigating the duality in regular polyhedra (conjugated polyhedra).

Students should have the opportunity to work with regular polyhedra built using different materials (wood, plastic, cardboard). You can find nets for these five shapes at www.anayaeducacion.es

Euler’s formula is very useful for checking whether we have counted the number of faces, vertices and edges of a polyhedron properly. If the result of f + v – e is not 2, we know that we have made a mistake, which is easy to do.

ANSWER KEY FOR ‘THINK AND PRACTISE’

The table shows clearly that the dodecahedron and the icosahedron are dual polyhedra and that the tetrahedron is dual of itself.

2 It is not semiregular because the same number of faces do not meet at all its vertices.

3 It is not semiregular because its lateral faces are not regular polygons.

4 Because the edges of a polyhedron are the edges of its faces. If all the faces are regular polygons, they will all have equal sides.

5 • Regular pentagonal prism: f = 7; e = 15; v = 10 8 7 + 10 – 15 = 2

• Antiprism: f = 14; e = 24; v = 12 8 14 + 12 – 24 = 2

6 a) f = 10; e = 20; v = 12 8 10 + 12 – 20 = 2 b) f = 7; e = 15; v = 10 8 7 + 10 – 15 = 2 c) f = 7; e = 12; v = 7 8 7 + 7 – 12 = 2

7 The polyhedron is not simple because it has a hole.

It does not fulfil Euler's formula: f = 16; e = 32; v = 16 8 16 + 16 – 32 = 0

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tetrahedron dodecahedron icosahedron faces 4 12 20 vertices 4 20 12 edges 6 30 30

Regular polyhedra polyhedron regular 1. Its faces are identical regular polygons. The same number of faces meet at each vertex. There are only five regular polyhedra: Although its six faces are identical equilateral triangles, this polyhedron not regular vertices and four faces meet at others. Duality in regular polyhedra If we join the centres of every two contiguous faces of cube with segment, we get an octahedron. If we do the same with an octahedron, we get square. We therefore say that the octahedron and the cube are dual polyhedra. The number of faces of polyhedron the same as the number of vertices of its dual. And they both have the same number of edges. The dodecahedron is dual of the icosahedron. And the tetrahedron dual of itself. Semiregular polyhedra A polyhedron semiregular when its faces are two or more kinds of regular polygons and the same polygons meet at each vertex. For example, these two polyhedra are semiregular: The one on the left regular pentagonal prism with square lateral faces. The one on the right is called an This one is regular hexagonal antiprism where one hexagon and three equilateral triangles meet at each vertex. Euler’s formula If we count the number of faces vertices and edges in simple polyhedron (with no holes), we get All the polyhedra that we usually work with are simple. This is why Euler’s For example, we can check that Euler’s formula is true for this pentagonal pyramid: 10 No of faces: No. of edges: ` a b b b → 10 REGULAR AND SEMIREGULAR POLYHEDRA 1 For many centuries, it was widely up of four elements: air, water, earth Plato associated regular polyhedron shape and so he said was the symbol of the The regular polyhedra are called Interesting A semiregular polyhedron must have all equal edges. It’s not difficult to see = 2 when he was 23 years born. However, we know by Euler’s faces, triangles faces, triangles 20 faces, triangles faces, squares faces, pentagons Copy and fill in the table on the right in your notebook with the number of faces, vertices and edges of the other three regular polyhedra. Check that the dodecahedron dual of itself. 2 We have already seen that this shape is not regular polyhedron. it semiregular? This truncated pyramid has square bases. Is it semiregular polyhedron? Why? Explain why the edges of semiregular polyhedron must all be equal. Check that Euler’s formula is true for the two semiregular polyhedra above. Check Euler’s formula for these polyhedra: Explain the reason why this polyhedron is not simple. Check that Euler’s formula a) b) c) Think and practise

students

‘Intuition

recommend that

use the

Truncating regular polyhedra

SUGGESTED METHODOLOGY

Here, we would like to reiterate the importance of students doing hands-on work with three-dimensional shapes so that they understand how these work. They should handle them, draw on them, cut them, etc. When working on this section, it is crucial that students look at regular polyhedra, draw on their faces, imagine cutting them, etc. This will help them to see how the semiregular polyhedra described here are formed.

This section ends with the study of other semiregular polyhedra, which students can construct from nets available at www.anayaeducacion.es . However, we would like to emphasise that the most important thing is for them to be able to imagine cutting regular polyhedra.

If possible, building regular polyhedra from plasticine, clay or other similar materials will help students who have less advanced spatial imaginations.

ANSWER KEY FOR ‘THINK AND PRACTISE’

1 a) Octahedron.

b) Yes, this is correct.

c) The shape that results from truncating the two dual polyhedra is the same.

2 x = 3 1 s, where s is the side of the triangle.

3 It has 8 faces, 4 regular hexagons and 4 equilateral triangles.

It has 12 vertices.

It has 18 edges, which measure one third of the edges of the tetrahedron.

4 It has 14 faces, 8 regular hexagons and 6 squares.

It has 24 vertices and 36 edges.

It is a semiregular polyhedron because two regular hexagons and one square meet at each vertex.

Euler’s formula: f = 14; e = 24; v = 36 8 14 + 24 – 36 = 2

5 • Truncated dodecahedron

It has 32 faces, 12 regular decagons and 20 equilateral triangles.

It has 60 vertices and 90 edges.

It is a semiregular polygon because two regular decagons and one equilateral triangle meet at each vertex.

Euler’s formula: f = 32; e = 60; v = 90 8 32 + 60 – 90 = 2

• Truncated icosahedron

It has 32 faces, 20 regular hexagons and 12 regular pentagons.

It has 60 vertices and 90 edges.

It is a semiregular polygon because two regular hexagons and one regular pentagon meet at each vertex.

Euler’s formula: f = 32; e = 60; v = 90 8 32 + 60 – 90 = 2

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Truncating means to cut off vertex of polyhedron with plane. We can get many new shapes by truncating the ones that we already know. Truncating at the midpoint of an edge Let’s look the shape we get from truncating all the vertices of cube with planes that pass through the midpoints of the adjacent edges We get shape called cuboctahedron with square faces (one for each face of the cube) and 8 triangular faces (one for each truncated vertex). Note that it is semiregular polyhedron. Similarly, we cut the vertices of regular dodecahedron with planes that pass through the midpoints of the adjacent edges, we get semiregular polyhedron called an icosidodecahedron. Truncating leaving part of the edges If we truncate the vertices of cube and leave part of the edges, the faces become octagons. If we cut at the right points, the octagons will be regular. This way, the is semiregular polyhedron. If we truncate cube at the right distance from its vertices, we get regular polyhedron called truncated cube, where two octagons and one triangle Similarly, the truncated tetrahedron, the truncated octahedron, the truncated dodecahedron and the truncated icosahedron are also semiregular polyhedra. Let’s look at those last two: Note that the truncated icosahedron has the same shape as some footballs. TRUNCATING REGULAR POLYHEDRA 2 Let’s think! The activity below will help you understand where the names come from. Archimedean solids Archimedes studied type of semiregular polyhedra known as Archimedean solids. There are 13 of them: the on this page, the truncated shapes on the previous page, and 6 that are more complicated. Now, we are going to truncate the other regular polyhedra by cutting through the midpoints of the adjacent edges. a) When we truncate tetrahedron this way, we get familiar shape. Which one? b) When we truncate an octahedron, we get the cuboctahedron from the top of the page. When we truncate an icosahedron, we get the icosidodecahedron above. Check it in these drawings. How do these results relate to the duality of polyhedra we learnt about in the previous section? 2 How far from the vertex do we have to cut the small triangles to make regular hexagon? How many faces does it have? How many of each kind? How many vertices and edges? What is the length of the edge in relation to the length of the original tetrahedron edge? Describe this truncated octahedron: Indicate the number of faces, edges, and vertices, and check that it semiregular polyhedron that fits Euler’s formula. Repeat the previous exercise for the truncated above. Think and practise Think and practise

GeoGebra

Draw three-dimensional shapes and trace their planes of symmetry.

Planes of symmetry of a shape

SUGGESTED METHODOLOGY

Students will be able to recognise planes of symmetry in three-dimensional shapes.

Two dual polyhedra, arranged according to their duality relationship, share the same planes of symmetry. Therefore, we can use one of them to help us find the planes of symmetry of the other.

ANSWER KEY FOR ‘THINK AND PRACTISE’

1 It must contain an edge and be perpendicular to two faces.

A tetrahedron has six planes of symmetry.

2 A regular hexagonal prism has six planes of symmetry, one for each axis of symmetry of its bases, and another parallel to the two bases.

3 Regular octahedron: it has two types of planes of symmetry, one that passes through four vertices and another that passes through two opposite vertices and the midpoints of two opposite edges.

Regular pentagonal pyramid: all of its planes of symmetry pass through its vertex and are perpendicular to the base and a lateral face.

Sphere: all of its planes of symmetry pass through its centre.

Be careful This not plane of symmetry of the cuboid. It does divide the shape into two equal parts, but they do not

OF SYMMETRY OF A SHAPE 3

two parts that reflect each other. Many three-dimensional PLANES

plane of symmetry of shape divides Any plane that contains the axis of the cylinder is plane of symmetry. them. It also has a plane of symmetry that is parallel to its bases.

Some shapes in nature, like slice of star fruit, have radial symmetry.

AXES OF ROTATION OF A SHAPE 4

Some shapes in nature, like star fruit cut in half, have radial symmetry. mathematical language, we say that has 5-fold axis of rotation A line is an -fold axis of rotation of shape if when you rotate the shape appears to return to its initial position times (including the initial position).

Axes of rotation of cube On the right, you can see 4-fold axes of rotation. When you rotate the cube around these axes, returns to its initial position four times. There are three of them. This is 2-fold axis of symmetry. them.

This 3-fold axis of symmetry. There are four of them. Axes of rotation of tetrahedron

UNIT Any two opposite edges make plane of symmetry. There are six of them. Planes of symmetry of prisms and cylinders Regular pentagonal prisms have planes of symmetry that contain the axes of symmetry of their bases and are perpendicular to them. They also have plane of symmetry parallel to their bases.

Think and practise

The axis passes through one vertex and the midpoint (centroid) of the opposite face. This 3-fold axis of same as the number of vertices).

What are the folds of the axes of rotation below? Does the prism have any other axis of rotation? And the pyramid?

This axis is perpendicular to two opposite edges at their midpoints. This 2-fold axis of rotation. There are three of them (one for every two edges).

Describe the axes of rotation of this octahedron: You can use the axes of rotation of the cube for help.

Axes of rotation of a shape

SUGGESTED METHODOLOGY

Here it is important to reiterate the importance of working with tangible three-dimensional shapes and using the dual shape in cases where this helps us to see the axes more easily.

When we say, for example, that ‘a cube has four 3-fold axes of rotation’, we do not expect students to memorise this fact, but rather to try and see it. To do this, it is crucial that they have access to the three-dimensional shape.

ANSWER KEY FOR ‘THINK AND PRACTISE’

1 The regular hexagonal prism has one 6-fold axis of rotation and six 2-fold axes of rotation.

The regular hexagonal pyramid only has a single 6-fold axis of rotation. It passes through the centre of the base and through the opposite vertex.

The circular sector is drawn with radius of We find the angle of the sector like we did above: 14 14

14.14cm 14.14

SURFACE AREA OF THREE-DIMENSIONAL SHAPES surface area of a polyhedron We get the surface area of polyhedron by adding up the areas of all its faces. We can look at the net of the polyhedron for help. surface of cylinder The lateral surface of cylinder rectangle whose base is equal to the perimeter of the circle, and whose height, h, is the height of the cylinder. Ar2h surface area of a cone The lateral surface of cone is the circular sector of circle with that intercepts an arc length that the same as the circumference of the base of the cone (2 ). To make net of the lateral surface area of the cone, we find the angle gg rg360 360 360 rg a A rg A 5 Find the total area of rightangled cone (height radius) with radius of 10 cm. What is the angle of the circular sector of its net? 10 cm 14.14cm 14.14cm 62 3 The generatrix rg A = 444 758 cm Perimeter of the base = 2 10 62.83 cm

Calculating the lateral area for trapezium. Remember, in trapezoid the area is equal to the semi-sum of the bases multiplied by the height. we now take the net of the lateral area as trapezoid with bases of 2

and 2 and height of Area This is the area we are looking for. BE CAREFUL! This not demonstration, just mnemonic rule. You can find demonstration at anayaeducacion.es surface area of sphere The surface area of sphere equal to the lateral area of the cylinder that encloses the sphere. And the same is true for the shapes we get from cutting spherical sections with parallel planes. A 2 π A = 2 h' These relationships between spheres and cylinders are very interesting. Even more interesting: if we draw any shape in the sphere and project it onto the cylinder, we get another shape that may look different but has the same surface area. 2 Archimedes. Sphere and cylinder The lateral area of cylinder equal to the area its inscribed sphere. And the total area of the cylinder? Well, we will see in the next section, the same relationship true for the volumes of these two figures: These relationships were discovered by Archimedes, and he was so proud that he asked that sphere inscribed

Surface area of three-dimensional shapes (I)

SUGGESTED METHODOLOGY

We highlight, in particular, the methods for using the Pythagorean theorem or similarity of triangles to find some basic elements for subsequent calculations (apothem of a pyramid, some elements of a truncated cone, the height of a spherical cap, etc.).

We also point out an often forgotten but simple and useful fact: the surface area of a segment or cap of height h is equal to that of the cylinder circumscribed in the corresponding sphere, with height h.

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plane of symmetry of shape divides it into two parts that reflect each other. Many three-dimensional shapes have plane of symmetry. Some have more than one. Planes of symmetry of cube Cubes have planes of symmetry that are parallel to its faces. There are three of them. Any two opposite edges make plane of symmetry. There are six of them. Planes of symmetry of prisms and cylinders Regular pentagonal prisms have planes of symmetry that contain the axes of symmetry of their bases and are perpendicular to them. They also have plane of symmetry parallel to their bases. Any plane that contains the axis of the cylinder is plane of symmetry. them. It also has a plane of symmetry that is parallel to its bases. PLANES OF SYMMETRY OF A SHAPE 3 This not plane of symmetry of the cuboid. It does divide the shape Explain the requirements for plane to be plane of symmetry of tetrahedron. Draw hexagonal prism like this in your notebook. Draw plane of symmetry that is perpendicular parallel to its bases. Describe some planes of symmetry for the octahedron, the regular pentagonal pyramid and the sphere. AXES OF ROTATION OF A SHAPE 4 star fruit, have radial symmetry. Some shapes in nature, like star fruit cut in half, have radial symmetry. mathematical language, we say that has 5-fold axis of rotation of shape if when you rotate the shape around appears to return to its initial position times (including the initial position). Axes of rotation of cube On the right, you can see 4-fold axes of rotation. When you rotate the cube around these axes, returns to its initial position four times. There are three of them. axis of symmetry. them. axis of symmetry. them. Axes of rotation of tetrahedron The axis passes through one vertex and the midpoint (centroid) of the opposite face. This 3-fold axis of rotation. There are four of them (the same as the number of vertices). This axis is perpendicular to two opposite edges at their midpoints. This 2-fold axis of rotation. There are three of them (one for every two edges). What are the folds of the axes of rotation below? Does the prism have any other axis of rotation? And the pyramid? Describe the axes of rotation of this octahedron: You can use the axes of rotation of the cube for help. Think and practise Draw three-dimensional shapes and trace their planes of symmetry.

it into

2 It has three 4-fold axes of rotation; six 2-fold axes of rotation and four 3-fold axes of rotation. shapes have plane of symmetry. Some have more than one. Planes of symmetry of cube Cubes have planes of symmetry that are parallel to its faces. There are three of them.

Explain the requirements for plane to be plane of symmetry of Draw hexagonal prism like this in your notebook. Draw plane of symmetry that is perpendicular to its bases and another one that parallel to its bases.

Describe some planes of symmetry for the octahedron, the regular pentagonal pyramid and the sphere.

This is a full review of the procedures for finding the surface areas of all the known threedimensional shapes.

surface area of truncated cone Relationship between radius, height and generatrix In truncated cone, the height, h, the difference between the radii, and the generatrix, form right-angled triangle. Therefore:

13 16

Calculate the area of these polyhedra that we get from cube with 12 cm edges: 12 6 12

REINFORCEMENT AND EXTENSION

We recommend:

• From the DIVERSITY AND INCLUSION section in the anayaeducacion.es resource bank:

Extension: Practice Exercise 1, worksheet B. Application Exercises 2 and 3, worksheet B.

15 20

a plane parallel to the base at height of 4 cm. First, we have to find the generatrix: We need to calculate the radius of the smaller base and the generatrix of the truncated cone We use similarity and the Pythagorean theorem: Calculate the surface area: A 3.14 (9 + 6) 235.5 cm A A 235.5 + 367.38 602.88 cm We cut sphere with a radius of 20 cm and get section that is circle with radius of 16 cm. What the area of the spherical cap? We calculate the height, of the cap:

We calculate the area of the cap:

Cooperative learning

Calculate the surface area of these shapes: 17cm 13 cm cm We have sphere with diameter of 30 cm. Calculate: a) The surface area of spherical segment with height of b) The surface area of spherical cap with base radius of a) right-angled prism whose base rhombus with diagonals of 12 cm and 20 cm, and lateral edges of b) right-angled pyramid with the same base and same lateral edges as the prism above. cuboctahedron with 10 cm edges. d) truncated dodecahedron with 10 cm edges.

Surface area of three-dimensional shapes (II)

17 cm surface area of the shape we get from dividing in half, and the total surface area of the truncated cone we get from cutting with section parallel to the base at height of 5 cm.

ANSWER KEY FOR ‘THINK AND PRACTISE’

1 A 8 Area = 792 cm2

B 8 Area = 635.64 cm2

C 8 Area = 772.73 cm2

D 8 Area = 619.02 cm2

2 A 8 Area = 812.64 cm2

B 8 Area = 437.76 cm2

You can use the ‘Cooperative learning: Questions bag’ resource to complete activity 7 on page 237, if you think it is appropriate.

3 A 8 Area = 678.58 cm2

B 8 Area = 366.06 cm2

C 8 Area = 452.39 cm2

4 A 8 Area = 1 856 cm2

B 8 Area = 1 570.8 cm2

5 A 8 Area = 742.42 cm2

B 8 Area = 531.21 cm2

C 8 Area = 551.22 cm2

6 a) Area = 565.49 cm2

b) Area = 565.49 cm2

7 a) Area = 1 359.6 cm2

b) Area = 235.9 cm2

c) Area = 946.41 cm2

d) Area = 7 260 cm2

SUGGESTED METHODOLOGY

This section is also a review. We remind students how to calculate the volumes of many of the three-dimensional shapes they know, focussing particularly on the more complex ones: truncated cones and spherical segments and caps.

AND EXTENSION

We recommend:

•

A 8 V = 864 cm3

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UNIT 11 SURFACE AREA OF THREE-DIMENSIONAL SHAPES 5 Problems solved regular hexagonal right-angled pyramid has an edge of the base of 10 cm and lateral edges of 13 cm. What is the total surface area? Calculate the apothem of the pyramid and the apothem of the base ( – –Calculate the surface area: A 61 61 66 259.8 cm 360 259.8 = 619.8 cm right-angled pyramid has a height of 15 cm and square base with 16 cm sides. What is the total surface area? cone has a height of 12 cm and base radius of 9 cm. Calculate the lateral area and total surface area of the truncated cone we get from cutting it with

–

What is the surface area of the prism and the pyramid? Both have regular hexagonal base. cm cm Calculate the surface area of these shapes: 12 cm C

Volume of three-dimensional shapes

239 238 UNIT volume of prisms and cylinders We get the volume of any shape with two equal bases parallel to each other (prismatic shape) by multiplying the area of the base by the height. volume of pyramids and cones The volume of pyramid or cone one-third the area of the base multiplied by the height. V A h volume of spheres The volume of a sphere with radius Note that the volume of sphere is two-thirds the volume of the cylinder that 3 2 (2 3 2 An interesting relationship: + V V V cm cm volume of spherical segment The relationship between the volumes of the hemisphere, the cone and the cylinder is also true for the sections we get from cutting them with parallel planes. V –VOLUME OF THREE-DIMENSIONAL SHAPES 6 Note, as we said the last section: 2 3 Remember The relation on the right allows us to calculate the volume of spherical segment by subtracting the volume of truncated cone from the volume of 30 We cut cone with base radius of 10 cm and height of 30 cm with plane parallel to its base at a height of 12 cm. Calculate the volume of the truncated cone. 4 Calculate the volume: 10 30 1 18 140 – 678.24 461.76 cm 2 We cut a sphere with a radius of 20 cm with two parallel planes that are 5 cm and 15 cm from the centre. Calculate the volume of the spherical segment between the two planes. V 20 10 = 4 000 3 V V Calculate the volume of these prisms that we get by cutting cube with 12 cm edges: Calculate the volume of these pyramids whose bases are regular polygons: cone and the truncated pyramid: A We cut sphere with diameter of 36 cm with two parallel planes. One passes through the centre and the other is 12 cm from the centre. 36 The sphere divided into three spherical segments. Calculate the volume of each one.

REINFORCEMENT

From the DIVERSITY AND INCLUSION section in the anayaeducacion.es resource bank: Extension: Practice Exercises 2 and 3, worksheet B. ANSWER KEY FOR ‘THINK AND PRACTISE’

8 V = 1 296 cm3

8 V = 864 cm3

2 A 8 V = 583.76 cm3

B 8 V = 703.53 cm3

3 A 8 V = 774.92 cm3

B 8 V = 640.8 cm3

4 Upper portion 8 V = 1 809.56 cm3

Middle portion 8 V = 10 404.95 cm3

Lower portion 8 V = 12 214.51 cm3

The terrestrial sphere The Earth is sphere that spins like top. Since we are rotating with it, surround us. The most important one when appears and disappears on the horizon: our days and nights.

Equator: aequus equal equator an ‘equitable’ plane that divides the sphere into two equal Because the Sun passes over the meridian noon.

Parallel 60° N

COORDINATES West

where there is an important astronomical observatory.

latitude of any point on Earth is the angular measurement of the meridian arc from that point to the equator. We have to say if it is north (N) or south (S) of the equator. Auckland is at the parallel 37° S. All the points on any parallel have the same latitude. longitude longitude of any point on the Earth is the measure of the angle formed by the plane of the point’s meridian with the plane of the Greenwich meridian. We also have to indicate whether is to the east (E) or west (W) of the Greenwich meridian. Auckland at the meridian 174° E. geographic coordinates longitude geographic coordinates of Auckland are 174° E 37° S, which is the antipode of Jerez de la Frontera, which has geographic coordinates of 6° W 37° N.

Enterprising culture

7.5°W

Greenwich meridian, 0°

The surface of the Earth is divided The Jerez meridian is the antimeridian of Auckland. So they are always 12 hours apart.

Calculate the length in kilometres of the parallels at the following latitudes: a) 60° b) The radius of the Earth is around 6 371 km.

Your turn Find the length of the parallel 45° in kilometres.

UNIT

Time zones When the Sun the meridian of location, we say that it noon. When it passes over its antimeridian, we say that it midnight. Therefore, will be noon at different time at each longitude. If clocks were set with this in mind, places that are near each other would have similar times but not the same. This could lead to chaos. That why we have hourly jumps. They are determined as follows: A spherical segment of 15° is created with its centre at the 0° meridian (360° 24 15° every hour). In that segment, it will be noon when the Sun passes the 0° meridian. The other 23 sections are formed the same way. Many regions do not follow the they should have according to their geographic location. In fact, mainland Spain’s location corresponds to the time most EU countries). Let’s suppose that every region uses its geographically correct time zone. For instance, it noon in London (longitude 0°), let’s see what time it would be in Istanbul (longitude 29° E). Look at the image on the left. Since London is in later, 14:00. If we wanted to know the time in city in time zone to the west of London, we would subtract the corresponding hours.

In the figure, we can see

Geographic coordinates

SUGGESTED METHODOLOGY

Here we discuss the terrestrial sphere, geographic coordinates and some of the consequences of the Earth’s rotation: day and night and time zones.

Although students study this content in relation to other subject areas, it will be useful for them to review and reinforce them from a mathematical standpoint.

The set task will help students to think about their personal skills, strengths and weaknesses when faced with difficult situations and new challenges.

We recommend working with globes, which will help students to visualise what they are learning. To help students understand geographic coordinates and time zones, we recommend using blank spheres, which students can use to draw different parallels, meridians, time zones, etc.

ANSWER KEY FOR ‘YOUR TURN’

The parallel at 45° is approximately 28 286.82 km.

ANSWER KEY FOR ‘THINK AND PRACTISE’

1 a) 6 370 km

b) 510 million km2 c) 3.25 billones km3 d) 21.25 million km2

2 a) 6 736.3 km b) 17 322 km

3 It is 7 in the afternoon in Hiroshima.

EXERCISES AND PROBLEMS

Exercises and problems solved

one? c) What are the axes of rotation of each polyhedron (the prism and its dual)? What the fold of each one?

following three-dimensional shapes: a) b) 13cm 13cm 15 cm 10 cm

13cm 10cm Calculate the surface area and volume of the following three-dimensional shapes: a) Prism with height of 20 cm and whose base rhombus with diagonals of 18 cm and 12 cm. b) Regular hexagonal pyramid with lateral edge of 18 cm and base edge of 6 cm. c) Regular octahedron with 10 cm edges. d) Cylinder with height of 27 cm and base Cone with radius of 9 cm and generatrix of 15 cm. f) Hemisphere with radius of 10 cm. g) Sphere inscribed in cylinder with height of 1 m. h) Spherical cap with height of 7 cm from a sphere 8 Find the surface area and volume of these three-dimensional shapes a) b) cm

Geographic coordinates m. in time zone zero, what time is three times zones to the east? And five time zones to 10 We know that it 9 o’clock in the morning in Dublin (longitude 6° W). Use this diagram to find what time is in Monterrey (100° W). A ‘nautical mile’ the distance between two points on the equator that are 1' longitude apart. What is the length of nautical mile?

SUGGESTED METHODOLOGY

The geometry activities are ideal for work in small groups, which will encourage peer learning (drawing, building, checking, solving, etc.). They are also suited for learning through discovery, making proposals and sharing conclusions.

We recommend completing the activities on this page using this methodology, if you think it appropriate.

We also suggest trying the following with the activities on this page, or wherever you think suitable:

• Present a solution to one of the solved activities containing errors and ask students to identify and correct them.

• Ask students to discuss and correct their work in pairs.

ANSWER KEY FOR ‘YOUR TURN’ 1 a) 41 833 km b) Approximately 216 683 330 km2.

Approximately 42.5 %.

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11 The Earth rotates around an imaginary line that passes through its centre and meets the surface at two points: the poles. The planes that contain the Earth’s axis cut the surface of the Earth into great circles called meridians. All meridians pass through the poles. The planes that are perpendicular to the circumferences called parallels. The one that equator. It is great circumference that divides the northern and southern hemispheres. Geographic coordinates Every point on the Earth (for instance, Auckland, New Zealand) has one parallel and one meridian that pass through it. The parallels and meridians are referenced according to their position from two specific great circles: The equator. The meridian that passes through Greenwich, city near London

GEOGRAPHIC

a) Calculate the radius of the Earth in kilometres. b) Calculate its surface area in square kilometres. c) d)

boat goes from point A latitude N and 10° longitude W, to another point B with the same latitude and 80° longitude E, along the parallel they share. a) How far does it travel? b) How far would it travel if the difference in longitude between the two points was 180°? In Rio de Janeiro (43° W) it 7 o’clock in the morning. What time is in Hiroshima (132° E)? Think and

that the radius of the Earth, equal to the side of an equilateral triangle. So the radius of the parallel 60°, The length of the parallel 60° is: 015 km b) In this figure, the radius, we need is equal to the height of an equilateral triangle. Using the Pythagorean theorem: /2) The length of the parallel 30° is: The metre, unit of measure, used to be defined as ten-millionth quadrant of meridian of the Earth. In other words, the distance around an Earth meridian is 40 000 000 metres. Based on this:

practise

Assessment

anayaeducacion.es In ‘My web resources’ there are documents to prepare a portfolio.

Exercises and problems

DO YOU KNOW THE BASICS?

Regular and semiregular polyhedra. Axes and planes of symmetry

1 Only the orange polyhedron is semiregular. The blue, yellow and green polyhedra do not have edges of equal length, so they are not semiregular. The purple polyhedron is regular (icosahedron).

2 Each of them has 5 planes of symmetry.

3 Each piece has 2 planes of symmetry.

4 All the planes of symmetry of the cube inscribed in the octahedron are also planes of symmetry of the octahedron.

5 a) The prism is semiregular, but its dual is not. b) The prism has seven planes of symmetry. Its dual has the same planes of symmetry. c) The prism has thirteen axis of rotation. Its dual has the same planes of symmetry.

Surface area and volume of three-dimensional shapes

6 a) A total = 1 066 cm2 V = 1 246 cm3 b) A total = 340 cm2 V = 363.67 cm3

7 a) A total = 1 081.6 cm2 V = 2 160 cm3 b) A total = 399.6 cm2 V = 505.128 cm3 c) A total = 346.4 cm2 V = 473.4 cm3 d) A total = 1 495 cm2 V = 4 156.38 cm3 e) A total = 678.59 cm2 V = 1 017.88 cm3 f ) A total = 628.32 cm2 V = 2 094.4 cm3 g) A total = 31 415.93 cm2 V = 523 598.78 cm3 h) A total = 527.79 cm2 V = 1 488.07 cm3

8 a) A total = 204.21 cm2 V = 150.8 cm3 b) A total = 265.32 cm2 V = 249.6 cm3

Geographic coordinates

9 Three time zones to the east it is 11:00 a. m. Five time zones to the west it is 3:00 a. m.

10 2 o’clock in the morning.

11 1 nautical mile = 1.85 km

TRAINING AND PRACTICE

EXERCISES AND PROBLEMS

What are the planes of symmetry of a cuboid with square base? And the axes of rotation? What is the fold of each one? Answer the same questions for cube. You already know the characteristics of a dodecahedron (faces, vertices). Now, describe 14 You already know the characteristics of an 15 Draw this square antiprism in your notebook:

How many planes of symmetry does have? three dimensions of cuboid? What fold are they?

We know that regular icosahedron has several planes of symmetry. For example, you look at two of its opposite faces, the three planes that pass through their three heights are planes of symmetry.

a) Are there also planes of symmetry that pass through its opposite edges? How many are there? b) How many planes of symmetry does have? c) We know that the axis of rotation that passes through two of its opposite vertices is 5-fold. What is the fold of those that pass through the midpoints of two opposite edges? And of the axes that pass through the centres of two opposite faces?

Find the surface area and volume of these shapes: a) b) cm

Calculate the surface areas and volumes of these three-dimensional shapes: a) b)

cm 10 cm We rotate right-angled triangle around its two legs, of 9 cm and 12 cm, and get two cones. Draw them in your notebook and find the surface area and volume of each one. 20 Calculate the surface area of the threedimensional shapes described bellow: a) regular pentagonal right-angled prism with edges measuring 10 cm. b) regular dodecahedron with edges of 10 cm.

Calculate the total area of these regular and semiregular polyhedra. The edges are all 8 cm:

We know that the sum of the surface areas of shapes A and F is triple the area of shape B. So, A + F = 3B Which of the following statements are true or false? a) b) Find the surface area and volume of these regular prisms. In both cases the basic edge is 10 cm and the height cm. a) b)

c) d) 15 m m 24 Find the surface area and volume of this regular 8cm

To find the height, H remember that AO h, where the height of one face. 25 The base of cuboid is 240 cm 44 cm. Its Calculate the diagonals of its faces and the main diagonal. Calculate the volume of this truncated pyramid with square bases: We cut sphere with radius of 24 cm with two parallel planes. One passes through the centre and areas and volumes of the three segments. We cut sphere with diameter of 50 cm with two parallel planes that are 8 cm and 15 cm from the centre. Find the volume of the spherical segment between the two planes.

29 Target 7.3. How much does it cost to insulate each roof knowing that the asphalt sheet is 9/m m m 30 Rome is one time zone

12 A cuboid with a square base has five planes of symmetry. It has one 4-fold axis of rotation and four 2-fold axes of rotation.

A cube has 9 planes of symmetry, three 4-fold axes of rotation, four 3-fold axes of rotation, and six 2-fold axes of rotation.

13 It has 32 faces, 12 regular decagons and 20 equilateral triangles. It has 60 vertices and 90 edges.

14 It has 32 faces, 20 regular hexagons and 12 regular pentagons. It has 60 vertices and 90 edges.

Linguistic plan

The ‘Descriptive text’ resource helps students improve their written expression when describing what an object is like.

SDG commitment Watch the video for target 7.3.

15 a) Four planes of symmetry.

b) It has one 4-fold axis of rotation.

16 It has three 2-fold axes of rotation.

17 a) Yes, there are 15 planes.

b) It has 15 planes of symmetry.

c) The axes of rotation that pass through the midpoints of two opposite edges are 2-fold, and the others are 3-fold.

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EXERCISES AND PROBLEMS SOLVED International Space Station The International Space Station (ISS) research centre that of 400 km. a) What is the distance to the farthest point on Earth that can be seen from the ISS? b) What is the total surface area of the Earth that can be seen from the ISS? What percentage of the Earth can be seen from the Radius of the Earth: According to the figure on the right, the distance one leg of right-angled triangle. We can find the distance using the Pythagorean theorem: 771 6 371 293 –k We have to find the surface area of the spherical cap. The surface area of spherical cap the same as the lateral area of the circumscribed cylinder, so we calculate to find the height of that cylinder: 371 6 771 5 995 km Therefore, the height of the circumscribed cylinder is Therefore: 371 376 15 000 000 km c) We calculate the surface area of the Earth and then, find the percentage that corresponds to the cap: The percentage corresponding to the cap is: Percentage 15000000 15 = 0.0294 ≈ % Answer the same questions for second-generation Meteosat satellite in orbit 36 000 km above the Earth. If one radiator needed to heat every 50 m of a house, how many radiators are needed in this one? We divide the house into several pieces and find the volume of each one. To do this, we have to calculate some measurements and ): 4 –b 42 5 – = 3.12 m 31 5 – = 1.87 m · (5.6 + 5.5) = 277.5 m = 4 · 5 · 1,6 = 32 m V = 15.6 + 277.5 + 55 + 32 = 380.1 m Since 380.1 : 50 = 7.602, radiators are needed. 371 771 771 4m 6.8m 4 6.8 5.6 Axes and planes of symmetry Which of these polyhedra are semiregular? If they are not, explain why: 2 A cube’s plane of symmetry divides in two equal cuboids. Indicate the planes of symmetry one of them has. Indicate the planes of symmetry of one of the pieces into which this cube is divided: Remember the duality between the cube and the octahedron (faces-vertices). Use the planes of symmetry of cube to describe the planes of symmetry of an Draw the dual polyhedron of the following regular hexagonal prism in your notebook: a) Are the prism or its dual semiregular polyhedra? b) What are the planes of symmetry of each one? c) What are the axes of rotation of each polyhedron (the prism and its dual)? What the fold of each one? following three-dimensional shapes: a) b) 13cm 13cm 15 cm 10 cm 13cm 10cm Calculate the surface area and volume of the following three-dimensional shapes: a) Prism with height of 20 cm and whose base rhombus with diagonals of 18 cm and 12 cm. b) Regular hexagonal pyramid with lateral edge of 18 cm and base edge of 6 cm. c) Regular octahedron with 10 cm edges. d) Cylinder with height of 27 cm and base Cone with radius of 9 cm and generatrix of 15 cm. f) Hemisphere with radius of 10 cm. g) Sphere inscribed in cylinder with height of 1 m. h) Spherical cap with height of 7 cm from a sphere 8 Find the surface area and volume of these three-dimensional shapes a) b) cm Geographic coordinates m. in time zone zero, what time is three times zones to the east? And five time zones to 10 We know that it 9 o’clock in the morning in Dublin (longitude 6° W). Use this diagram to find what time is in Monterrey (100° W). A

mile’ the distance between two points on the equator that are 1' longitude apart. What is the length of nautical mile? EXERCISES AND PROBLEMS UNIT

‘nautical

to the east of zero, and New York is five time zones to the west. If plane takes off from Rome 11 p.m. and the flight is 8 hours, what time will it be in New York when lands? It is p.m. in Havana (82° W). Use times zones to say what time in each city. Maputo (Mozambique) 2 p. Natal (Brazil) 3 a. Astana (Kazakhstan) 8 p. Temuco (Chile) 0 a. Honolulu (Hawaii) 11 a. Dakar (Senegal) Kathmandu (Nepal) 6 a. Melbourne (Australia) 32 Two cities have the same longitude, 15° E, and the distance between them? 33 A plane has to go from point to point B, which are diametrically opposed on the parallel 45°. The plane can either follow the parallel APB or the polar route ANB What the distance it would the same latitude: 30° N. Their longitudes are 30° E, 90° W and 95° W. How far would plane travel from Alexandria to New Orleans along the parallel 30° N? And from Alexandria to Houston?

18 a) A total = 703.72 cm2

V = 760.27 cm3

b) A total = 5 864.31 cm2 V = 30 717.79 cm3

19 Rotation around the leg of 12 cm 8 A total = 678.59 cm2 V = 1 018 cm3 Rotation around the leg of 9 cm 8 A total = 1 017.88 cm2 V = 1 357 cm3

20 a) A = 844 cm2 b) A = 2 064 cm2

21 Shape A 8 A a = 110.88 cm2

Shape B 8 A b = 221.76 cm2

Shape C 8 A c = 384 cm2

Shape D 8 A d = 605.76 cm2

Shape E 8 A e = 1 714.56 cm2

Shape F 8 A f = 554.4 cm2

Shape G 8 A g = 1 373.76 cm2

All the statements are true.

22 a) A total = 744 cm2 V = 1 376 cm3 b) A total = 1 122.8 cm2 V = 3 862.4 cm3

23 a) A total = 205.74 m2 V = 207.35 m3

b) A total = 640.88 m2 V = 565.49 m3 c) A total = 1 644.5 m2 V = 4 452 m3

d) A total = 71.05 m2 V = 117.81 m3

24 A total = 110.88 cm2 V = 60.34 cm3

25 The diagonals of its faces measure 26.7 dm and 12.5 dm. The main diagonal measures 27.06 dm.

26 V = 1 293.3 m3

27 V portion 1 = 24 663.61 cm3

V portion 3 = 28 952.92 cm3

28 V = 36 651.9 cm3

V portion 2 = 4 289.31 cm3

29 Insulating roof A costs about €281 and roof B about €425.

30 It will be 1:00 a.m. (of the next day) in New York.

31 Maputo (32° E) 8 3 a. m. Natal 8 11 p. m. Astana (71° E) 8 6 a. m.

Temuco (73° O) 8 8 p. m.

Honolulu (158° O) 8 2 p. m. Dakar (16° O) 8 0 a. m. Kathmandu (85° E) 8 7 a. m. Melbourne (144° E) 8 11 a. m.

32 6 670.65 km

33 Following the parallel it would travel 14 153 km, and following the polar route, 10 008 km.

The polar route is shorter.

34 From Alexandria to New Orleans a plane would travel 5 724.33 km. From Alexandria to Houston it would travel 6 201.36 km.

SOLVE SIMPLE PROBLEMS

35 V = 225.8 cm3

36 We get a bigger volume by connecting the 20 cm sides.

37 The height is 35.36 cm.

38 V = 80 cm3

39 The cone has a bigger volume.

194

40 a) Faces: 20; Vertices: 12

b) Hexagons: 20; pentagons: 12

c) A = 7 260 cm2

41 It would not be entirely submerged.

42 The capacity of the basin is 880 690 L and the amount of water used in 10 h is 900 000 L. He cannot water the fields for 10 hours without topping up the basin.

43 The price should be €2.50.

YOU CAN ALSO DO THIS

44 a) V = 1 307.24 cm3 b) h = 6.67 cm

45 There are 240.48 cm3 = 0.24048 L left in the bowl.

46 Shape A weighs 21 kg, and shape B weighs 18 kg.

47 A = 468.1 cm2 V = 504 cm3 48 a) r = 1/2 b) V tetrahedron = 1.84 cm3; V octahedron = 7.37 cm3

THINK A LITTLE MORE

49 From the plane, 0.078 % of the Earth's surface can be seen.

50 Shape A weighs 17 kg, shape B, 16 kg and shape C, 8 kg.

51 A truncated pyramid = 84.3 cm2

V truncated pyramid = 43.5 cm3 A prism = 47.7 cm2 V prism = 18.8 cm3

DID YOU UNDERSTAND IT? THINK

52 a) T b) F c) T d) F

53 The cuboctahedron has the same planes of symmetry and the same axes of rotation as the cube.

54 If we reduce the radius by half, the volume is reduced to a quarter. If we reduce the height by half, the volume is reduced by half. 55 V V 8 1 SMALL

LARGE =

56 a) Euler’s formula: f = 14; e = 36; v = 24 8 14 + 24 – 36 = 2

b) The number of faces is reduced from 14 to 10; the edges are reduced from 36 to 24, and the vertices are reduced from 24 to 16.

c) Yes, Euler’s formula is true even though it has a hole: 10 + 16 – 24 = 2. This is the case because B is not a polyhedron, since two of its faces are not polygons (the upper and lower faces) as they have holes.

Maths workshop

READ, IMAGINE AND UNDERSTAND

Some ways of filling space

You help students visualise these divisions of space by taking a hands-on approach to the activity, asking a group of students to build various truncated cubes and the corresponding octahedrons, giving them the necessary nets to help them.

195

UNIT READ, IMAGINE AND UNDERSTAND Some ways of filling space When products are packaged, transported, and distributed, is important to make good use of space. We can think about the shapes of containers that do not leave gaps or empty spaces between them when piled. The best box is obviously one shaped as cuboid, especially the most regular We can also discover other ways of filling space with regular or semiregular polyhedra. These are not very practical in industry but can be beautiful and interesting in geometry. Suppose we cut the corners of cube to make equilateral triangle sections and regular octagonal faces. Note that: — When we pile cubes, eight meet at each vertex. When we cut them, the eight corners that meet vertex form regular octahedron. So we have found way to fill the space with truncated cubes and regular If we cut bigger sections so that the octahedrons touch at the vertices, the rest of the space is filled with semiregular polyhedron that you have already studied. MATHS WORKSHOP SELF-ASSESSMENT Explain why each of the following polyhedra is not regular: 2 Describe the polyhedron you get from truncating a regular octahedron with planes that cut the edges at one-third of their length. Is semiregular polyhedron? Explain why. Describe the planes of symmetry of this regular its axes of rotation and what fold are they? a) A pyramid with square base in which both the lateral edge and base edge are the same and measure 10 cm. b) and generatrix of 5 m. We cut sphere with radius of 8 cm with two parallel planes that are 2 cm and 3 cm from either side of the centre. Calculate the surface area of the spherical section between the two planes. 6 Calculate the volume of these geometric shapes: cm cm cm Calculate the volume of these shapes. cm 8.4 cm 6cm When it 8.00 a.m in the 0 zone, what time it in Two cities are on the equator and the difference between their longitudes is 10°. What is the distance (Radius of the Earth: 371 km) 10 The geographic coordinates of Saint Petersburg are Find the length of the parallel running between them. anayaeducacion.es Answer key. Look back at the content you have worked on and plan solutions to the problems that you may find. To do this, download the corresponding rubric from anayaeducacion.es think individually and share in groups. Now you can do the skills self-evaluation included in anayaeducacion.es. TEST YOUR SKILLS A

SELF-ASSESSMENT

1 Polyhedron A is clearly not regular because all its faces are isosceles triangles, not equilateral.

All of polyhedron B’s faces are equilateral triangles. However, there are vertices where 4 faces meet and others where 5 meet, so it is not regular.

Polyhedron C is not regular because, although all its faces are squares, in some vertices there are 6 faces, and in others there are 3 faces.

Polyhedron A: f = 8; e = 12; v = 6 → f + v – e = 8 + 6 – 12 = 2

Polyhedron B: f = 10; e = 15; v = 7 → f + v – e = 10 + 7 – 15 = 2

Polyhedron C: f = 30; e = 60; v = 32 → f + v – e = 30 + 32 – 60 = 2

2 The new shape has 6 squares and 8 regular hexagons. One square and two hexagons meet at each vertex. It is semiregular.

3 It has 9 planes of symmetry.

Axes of rotation: Three 4-fold, six 2-fold and four 3-fold.

4 a) A = 273.21 cm2 b) A = 602.88 cm2

5 A = 251.33 cm2

6 A a = 80 m2 A b = 116π = 364 cm2 A c = 216 cm2

V a = 48 m3 V b = 586.43 m3 V c = 224 m3

7 V a = 108 cm3 V b = 2 771 cm3 V c = 741 cm3

8 Three time zones to the east (3° E) it is 11:00 a. m.

Five time zones to the west (5° W) it is 3:00 a. m.

9 x 40 000 360 10 = 8 x ≈ 1 111

The distance between the cities is approximately 1 111 km.

10 There is an arc of 30° – 11° = 19° between the two cities. Therefore, the distance between them is 360 20 015 ° · 19° ≈ 1 056.35 km

CHALLENGES THAT LEAVE AN IMPRINT

THINK

In this unit, your students have made further progress in the preparation of the learning situation proposed in block 4. In particular, they have learned to calculate the volume of a building as well as the surface of a roof, which will help them to obtain the total amount of the renovation budget for the space they decide on.

Your students have a questionnaire at anayaeducacion.es which will help them to reflect on their own performance in the tasks proposed in this unit. It is a good idea to review as a group those aspects in which the students themselves have detected room for improvement.

TEST YOUR SKILLS

Your students have, also at anayaeducacion.es, a test that will help them to assess their level of acquisition of the skills put into play during the completion of the proposed ‘Challenge’.

196

UNIT READ, IMAGINE AND UNDERSTAND Some ways of filling space When products are packaged, transported, and distributed, is important to make good use of space. We can think about the shapes of containers that do not leave gaps or empty spaces between them when piled. The best box is obviously one shaped as cuboid, especially the most regular We can also discover other ways of filling space with regular or semiregular polyhedra. These are not very practical in industry but can be beautiful and interesting in geometry. Suppose we cut the corners of cube to make equilateral triangle sections and regular octagonal faces. Note that: — When we pile cubes, eight meet at each vertex. When we cut them, the eight corners that meet vertex form regular octahedron. So we have found way to fill the space with truncated cubes and regular If we cut bigger sections so that the octahedrons touch at the vertices, the rest of the space is filled with semiregular polyhedron that you have already studied. MATHS WORKSHOP SELF-ASSESSMENT Explain why each of the following polyhedra is not regular: Check Euler's formula holds for each one. 2 Describe the polyhedron you get from truncating a regular octahedron with planes that cut the edges at one-third of their length. Is semiregular polyhedron? Explain why. Describe the planes of symmetry of this regular its axes of rotation and what fold are they? a) A pyramid with square base in which both the lateral edge and base edge are the same and measure 10 cm. and generatrix of 5 m. 5 We cut sphere with radius of 8 cm with two parallel planes that are 2 cm and 3 cm from either side of the centre. Calculate the surface area of the spherical section between the two planes. Calculate the volume of these geometric shapes: cm cm cm Calculate the volume of these shapes. cm 8.4 cm 6cm When it 8.00 a.m in the 0 zone, what time it in Two cities are on the equator and the difference between their longitudes is 10°. What is the distance (Radius of the Earth: 371 km) 10 The geographic coordinates of Saint Petersburg are Find the length of the parallel running between them. anayaeducacion.es Answer key. Look back at the content you have worked on and plan solutions to the problems that you may find. To do this, download the corresponding rubric from anayaeducacion.es think individually and share in groups. Now you can do the skills self-evaluation included in anayaeducacion.es. TEST YOUR SKILLS ICT anayaeducacion.es Answer key for the self-assessment.

197 Notes