F
Level F
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2
Volume Volume is the amount of space a 3-dimensional (3D) shape takes up. To calculate a volume of a cuboid or cube, using the following formula.
height (h)
h l
b
length (l)
breadth (b)
Volume of cube or cuboid = l x b x h
Calculate the volume for the cubes and cuboids. Make sure you include the units. The first one has been done for you. The 3D shapes are not drawn to scale. c a b 4cm
2cm
6m
3cm
5cm
3cm
3cm
9m
1m
V=lxbxh = 5cm x 2cm x 4cm = 40 cmÂł e
d
f 6m
2.5m
2m
6.5m
2cm 9.12m
2cm
Level F
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The shapes below are all prisms.
l
l triangular prism
l parallelepiped
trapezoid
The shapes in grey are called cross section, so for the three shapes respectively, the shapes for each cross sections are: triangle, trapezium and parallelogram. Volume of prism = area of cross section x l
Calculate the volume for the prisms. Make sure you include the units. The first one has been done for you. The 3D shapes are not drawn to scale. a
c
b
3cm
2cm 5cm
7.5cm
6m
4cm 8cm
11m 8cm
5.2m
V = 2cm x 5cm x 7.5cm = 75 cmÂł
e
d 4mm
7mm 12mm 9mm
f 4cm 2.7cm
3cm
9cm
15cm 9cm
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/5 Week 37
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It is possible to find the missing length if we are given the volume and 2 of the sides. Have a look at the example below.
Example
The volume of the cuboid below is 120cm³. Find the length.
4cm l
6cm 1
Multiply the lengths that we know. 4cm x 6cm = 24 cm²
2
Divide 120cm³ by 24cm² to give you the missing length. 120cm³ ÷ 24cm² = 5cm
The missing length is 5cm. Using the method above or otherwise, find the missing letter of the following cuboids. c a b 9mm 4cm
1m
l
w
4cm Volume = 144cm³ V = 144cm³ 4cm x 4cm = 16cm² l = 144cm³ ÷ 16cm² = 9cm
l
4.6cm
5cm Volume = 11.5cm³
w
Volume = 22m³
e
d
15mm
2.2m
Volume = 810 mm³
h 6m 4.5m Volume = 243m³
f 5.1cm
8cm
l Volume = 326.4cm³
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Find the missing lengths (l) of the prisms. a
b l
l Area of parallelogram: 23cm² Volume: 126.5cm³
c
Area of trapezium: 100cm² Volume: 253.1cm³
d
6cm
12cm
17m
5m l Volume: 263.5cm³
l Volume: 144cm³
Answer the following questions. a
The volume of a cuboid is 120cm³. The length, width and height are consecutive numbers. What are the three numbers?
b
The volume of a cube is 8cm³. What is its length?
c
The volume of a triangular prism is 90cm³. It is 5cm long. The height and base of the triangle are the same. What is the height of the triangle?
d
A cube is cm³. Give
2 cm high. Find the volume of the cube in 3 your answer as a fraction.
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/8 Week 37
6
Surface Area To find the surface area of a cube, first look at what the cube looks like when it is opened.
A cube has 6 square faces. To calculate the surface area, calculate the area of one square then multiply it by 6 as there are 6 squares.
Example
Find the surface area of a cube that is 3cm long. The area of 1 square would be 9cm². So the surface area would be 9cm² x 6 since there are 6 faces in a cube. The surface area would be 54cm²
Calculate the surface area of the cubes if the lengths are given. The first one has been done for you. 12cm c b a 5cm 1.5cm
5cm x 5cm = 25cm² 25cm² x 6 = 150cm²
d
5.5cm
e
2.6cm
f
3.9cm
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7
Example
Calculate the surface area of a cuboid that is 5cm long, 10cm high and 2cm wide. 2
3 10cm
1
3 2cm
1
2
5cm
2
1
3
For each face (labelled 1 to 3), there are 2 of each. To find out the surface area, find out the area of 1, 2, 3 (all rectangles). Area of 1: 5cm x 10cm = 50cm² Area of 2: 2cm x 5cm = 10cm² Area of 3: 2cm x 10cm = 20cm² Then you add the areas together, then you double to get your surface area. 50cm² + 10cm² + 20cm² = 80cm² 80cm² x 2 = 160cm² The surface area is 160cm². Calculate the surface area of the cuboids. c
b
a
2m
22mm
10mm
15mm
25cm
5cm
4cm
2.5m
1m
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/3 Week 37
8
Calculate the surface area of the following shapes. The nets have been drawn for you. Drawings are not to scale. a
b
7cm 4cm
4cm
4cm 6.5cm
15cm 15cm triangular based pyramid
triangular prism
d
c 3.2cm 3cm
12cm
5cm
5cm
5cm
4cm
square based pyramid triangular prism
Level F
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9
For each 3D shape, calculate the surface area. The drawings are not to scale. a
cube
b
1cm
4cm
cuboid
12cm 5cm
c
triangular prism (isosceles) 13cm (side of triangle)
12cm (height) 10cm
e
d
square based pyramid
12cm (height of triangle)
20cm
cuboid
14cm
f
14cm
right angled triangular prism 5cm 4cm
3.5cm 2cm
4cm
3cm
3cm
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/6 Week 37
10
Problem Solving Find the volume of each shape in the boxes below. Give your answer in m³. For the second box, convert the centimetres into metres before finding the volume.
a
1
2
0.5m
5m
600cm
1m
3
b
c
1500cm 50cm
Find the difference in volume between the two shapes.
Tanya wants to wrap a present. The present is a cube with sides of 10cm. 1
How many cm² of wrapping up paper does she need?
2
If the wrapping up paper costs £0.05 per square centimetre (cm²). How much will it cost to wrap up the present?
A swimming pool is 50m long, 25m wide and 20m deep. Answer the following questions. 1
What is the volume of the swimming pool?
2
How many litres of water is needed to cover the swimming pool? (1000 litres = 1m³)
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d
Isaac is building a 3D model of a house for a school project. It is made using a cube and triangular prism. triangular prism 38cm
cube 20cm
20cm
1
What is the volume of the triangular prism?
2
What is the volume of the whole shape?
e
A mini rectangular pool is 100cm long, 150cm wide and 10cm high. 1
What is the volume of the mini rectangular pool?
2
If the Walls family had 100l of water, how much more water do they need to fill up the pool? (1 litre = 1000cmÂł) www.oxbridgeuk.com
/4 Week 37
12
Circle the net that does not make a 3D shape specified in each question. a
Cube
b
Triangular Prism
c
Tetrahedron (triangular based pyramid)
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/3
ISBN 978-1-78395-386-8
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F
Level F
Problem Solving
Name Class www.oxbridgeuk.com COPYRIGHT ACT
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14 1
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The volume of a cube is 343cm3. What is the length of this cube?
A cube’s surface area is 150cm2.
2
What is the length of the cube?
3
What is the volume of the cube?
Advanced Problem Solving
15
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This is an isosceles triangular prism. Using the diagram below, answer the following questions.
B 12cm
8cm
Acm
4
The area of the triangle is 24cm2. Find the value of A.
5
Find the volume of this prism.
6
Rectangle B has an area of 84cm2. What is the surface area of this prism?
7
If the value of A is increased by 1cm, how much would the volume increase? Advanced Problem Solving
16
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This is an isosceles trapezoidal prism. Using the diagram below, answer the following questions.
3cm 4m
5cm
9cm
7cm
8
What is the area of the trazpezium?
9
What is the surface area of this prism?
10
What is the volume of this prism?
11
One of a cube’s face has an area of 25cm2. What would be the difference in volume between that cube and this prism?
Advanced Problem Solving