NAME: SAYANI ROY ROLL NO: T-099 SUBJECT: MATHEMATICS TOPIC: MEASUREMENT OF VARIOUS SOLIDS CLASS: X SCOTTISH CHURCH COLLEGE (B.ED DEPARTMENT)
CONTENT:-
1. RIGHT PRISM 2. RIGHT CIRCULAR CYLINDER 3. RIGHT PYRAMID 4. RIGHT CIRCULAR CONE 5. SPHERE 6. HEMISPHERE
Exploring Solids • Polyhedron: a solid that is bounded by polygons, called faces, that encolose a single region of space. • Edge: the line segment formed by the intersection of two faces. • Vertex: a point where 3 or more edges meet. • Regular: all faces are congruent regular polygons
Types of Solids 1. Prism
polyhedron
2. Pyramid
polyhedron
3. Cone
not a polyhedron
4. Cylinder
not a polyhedron
5. Sphere
not a polyhedron
6. Hemisphere
not a polyhedron
1. RIGHT PRISM: A prism refers to a solid geometric figure whose two end faces are similar, equal and parallel rectilinear figures and whose sides are parallelograms.
some examples of prism
Right Prism Right Prism is a prism that has two bases, one directly above the other, and that has its lateral faces as rectangles. In a right prism, the edges of the lateral faces are perpendicular to the bases.
The Different Types of Prisms
There are as many different types of prisms, as there are polygons! A prism is named after the shape of its front and back face. A list of most commonly studied prisms at the elementary level,are given below. Triangular Prism
Square Prism
Rectangular Prism
Pentagonal Prism
Hexagonal Prism
Octagonal Prism
Parts Of A Right Prism
BASE: A prism is a polyhedron consisting of two parallel, congruent faces called bases. HEIGHT: The prism's height is the distance between its two bases LATERAL FACE: the faces which are used to join the bases of a Solid. These are parallelograms formed by connecting the corresponding vertices of the bases. LATERAL EDGES:. The segments connecting these vertices are lateral edges.
LATERAL AREA, SURFACE AREA AND VOLUME OF A RIGHT PRISMLateral Area: L.A = ph (p = perimeter of the base, h = height of prism) Surface Area: S.A= ph + 2B ( B = base area) = [Lateral Area + 2 (area of the base)] Volume: V = Bh
Example: Find out the lateral area, surface area and volume of a rectangular prism. given: a =10cm ,b=6cm & h =16cm
Answer:
sides of the rectangle, a=10cm b=6cm perimeter (p)= 2(10+6)cm =32 cm Area (B) =(10x6) cm2 =60 cm2
Now, lateral Area of the prism L.A = ph = (32X16) cm2 =512 cm2 Surface Area of the prism S.A= ph + 2B =512+ (2X60) =632 cm2 Volume of the prism V = Bh = (60X16) = 960cm3
2. RIGHT CIRCULAR CYLINDER: A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity. A cylinder generated by the revolution of a rectangle about one of its sides is called RIGHT CIRCULAR CYLINDER.
Parts Of A Right Circular Cylinder
BASE: A cylinder is a geometric solid that is very common in everyday life, such as a soup can. If you take it apart you find it has two ends, called bases, that are usually circular. The bases are always congruentand parallel to each other. If you were to 'unroll' the cylinder you would find the the side is actually a rectangle when flattened out. HEIGHT: T he height h is the perpendicular distance between the bases
RADIUS: The radius r of a cylinder is the radius of a base. AXIS: A line joining the center of each base.
SURFACE AREA,TOTAL ,TOTAL SURFACE AREA AND VOLUME OF A CYLINDER
The area of the rectangular sheet gives us the curved surface area of the cylinder. The length of the sheet is equal to the circumference of the circular base which is equal to 2 r. So, Surface of the cylinder,L cylinder = area of the rectangular sheet = length x breadth = perimeter of the base of the cylinder x h height =2rh
The total surface area, T =2 X area of the base + surface area = 2rh + 2r²
Volume of a cylinder, V = area of the base X height, = r²h
Example: The diameter of the base of a right circular cylinder is 7 cm and its height is 40 cm. Find the total surface area and volume of the cylinder?
Answer: Here diameter = 7 cm. Radius (r ) = 7/2 cm.
Height (h) = 40 cm. So the total surface area of cylinder, T =2rh + 2r² = (2x22/7x7/2x40)+(2x22/7x7/2x7/2) =957cm2
Volume of cylinder, V = π r2h V = 22/7x (7/2)2x 40 V = 22/7x 7/2x7/2x 40 V = 1540 cm3
3. RIGHT PYRAMID: In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle.
A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges.
Right pyramid : the apex of the pyramid is directly above the center of its base
Regular pyramid: the base of this pyramid is a regular polygon
Types Of PyramidsThere are many types of pyramids. Most often, they are named after the type of base they have. Let's look at some common types of pyramids below.
Pyramids with regular polygon facesfaces Pyramids
Tetrahedron
Square pyramid Pentagonal pyramid Hexagonal pyramid
PARTS OF A PYRAMID-
The perimeter is the distance around the base of the pyramid The height is the distance between the apex of a pyramid and the center of its base. The slant height is the diagonal height from the center of one of the base edges to the apex.
LATERAL AREA, SURFACE AREA &VOLUME OF A REGULAR PYRAMIDLateral Area: L.A. = 1/2 lp slant height)
(p = perimeter of base, l =
Surface Area: S.A. = L.A + B (B = area of base) =1/2 lp + B Volume: V = 1/3 Bh
( B = area of base, h = height)
Example: Find out the lateral area, surface area and volume of a square pyramid. given: a=12 cm, h=5cm, l=8 cm
Answer: Side of the square(a)=12cm Perimeter(p)= 4 a = (4 X 12) cm =48cm Area(B) = a2 =122 =144cm2 Now, lateral area of the square pyramid L.A. = 1/2 lp = 1/2 X 8 X 48 =192cm2
Surface area of the square pyramid S.A. = 1/2 lp + B = (192+144) cm2 Volume of the square pyramid V = 1/3 Bh =1/3 X 144 X 5 =240 cm3
4. RIGHT CIRCULAR CONE: A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. In common usage in elementary geometry, cones are assumed to be right circular, where circular means that the base is a circle and right means that the axis passes through the centre of the base at right angles to its plane.
A right circular cone is a cone:
whose base is a circle
in which there is a line perpendicular to the base through its center which passes through the apex of the cone.
which is made by having a right-angled triangle turning along one of the sides that form the right angle.
When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone. And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be right-angled.
Parts Of Right Circular Cone
Base: radius.
The base of the cone is a circle with O as center and OB or OC as
Axis : The axis of the cone is the straight line which remains fixed and about which the triangle is turned.
In the above right circular cone, the point A is the vertex of the cone. O be the center of the base of cone & AO be the AXIS of the cone.
Height & Slant Height: Axis AO which is perpendicular to the circulae base of the cone is called the HEIGHT (h). So, the length AO=h AC or AB is the SLANT HEIGHT (l) So,the length AC or AB =l
CURVED SURFACE AREA,TOTAL SURFACE AREA AND VOLUME OF CONE :-
Curved surface area of the cone:1/2 X perimeter of the base X slant height = 1/2 X 2πr X l = πr l Total surface area of the cone :Curved surface area of cone + Area of the circular base = πrl + πr² = πr( l + r ) Volume of the cone:1/3 X area of the circular base X height =1/3 X πr² X h =1/3πr²h
Example: Calculate the curved surface area,total surface area and volume of a right circular cone.Given that: r=6cm,h=8cm,l=10cm
Answer:
Curved surface area of a cone= πr l = (22/7) X 6 X 1O =188.57 cm2
Total surface area of the cone = πrl + πr² = (22/7) X62 +188.57 =113.14+188.57 =301.71 cm2
Volume of the cone = 1/3πr²h =1/3 X (22/7) X 62 X 8 =301.71 cm3
5. SPHERE: In space, the set of all points that are a given distance from a given point, called the center. A sphere is formed by revolving a circle about its diameter
Spheres – special segments & lines
Radius: A segment whose endpoints are the center of the sphere and a point on the sphere. Diameter: A chord that contains the sphere’s center Chord: A segment whose endpoints are on the sphere
Notice these interesting things-
It is perfectly symmetrical It has no edges or vertices(corners) It is not a polyhedron
Surface Area & Volume of Sphere
Surface Area (SA) = 4 π r2 Volume (V) =4/3 π r3
Example 1: Find the surface area of a sphere whose diameter is 21cm
Answer: Diameter=21cm Radius(r) = (21/2) cm SA=4 π (21/2)2 = 4 X (22/7) X (21/2)2 =1386 cm2
Example 2: Find the volume of a sphere .Given that r=21cm Answer: we know that, V=4/3 π r3 =4/3 x (22/7) x (21)3 =38808cm3
6.HEMISPHERE : It is an exact half of a sphere
When a plane cuts across a sphere at its center, it forms two hemispheres. The upper and lower parts of the sphere are equal halves
Surface area of hemisphere There are two types of surface areas of the hemisphere total surface area and curved surface area.
The curved surface area is the area of the outer surface of the hemisphere. Since, hemisphere is half of sphereTherefore, Curved SA = (4 π r2)/2 =2 π r2
While the total surface area includes the area of the curved surface and area of the upper circle. Total SA =Curved SA +Area of upper circle = (2 π r2 + π r2)
Example 1: Find the total surface area of a Hemisphere, having radius of 10 cm
Answer: Total SA= (2 π r2 + π r2) =3 π r2 =3x π x102 =300 π cm2
Volume of hemisphere Volume of a hemisphere is half of the volume of a sphere. V= (4/3 π r3)/2 =2/3 π r3
Example 2: Find the volume of a Hemisphere.Given r=6cm Answer:
r=6cm
We know that, V=2/3 π r3 = (2/3) X π X 63 =144 π cm3
EXERCISE 1. Find out the lateral surface area and volume of a rectangular prism whose sides are a=10cm, b=4cm & h=20cm.
2. Find the volume of this pentagonal prism if the area of the base is 60cm².
3. If the diameter of the base of a right circular cylinder is 14cm and height is 20cm, then find the total surface area and volume of the cylinder.
4. A cylindrical pillar is 50 m in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs 12.50 per m².
5. Find the total surface area and volume of a pyramid whose base is a square with sides 10cm each and height is 5cm and slant height is 6 cm.
6. Find the ratio of curved surface area and total surface area of a right circular cone whose base is of radius r cm, height is h cm and slant height is l cm.
7. Diameter of the base of a ice cream cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.
8. Find the volume of a sphere whose radius is 10cm & hence find the volume of hemisphere of same radius.
9. Find the radius of a sphere whose surface area is 154 cm².
10. What is the ratio of the volume of a hemisphere & a sphere of same radius?
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