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5.08 Finding dimensions from surface area or volume

Worksheet
Work backwards from area and surface area
1

Each face of a cube has an area of 64\text{ cm}^{2}.

a

Find the length of the cube's edge.

b

Find the volume of the cube.

2

Find the side length of a cube that has a surface area of 600\text{ cm}^2.

3

The curved area of a cone is 380\text{ cm}^{2} and the slant height is 15\text{ cm}:

a

Find the radius of the cone, correct to two decimal places.

b

Find the perpendicular height of the cone, correct to one decimal place.

c

Find the total surface area of the cone, correct to the nearest whole number.

4

Find the height, h, of this closed cylinder if its surface area is 27\,288\text{ mm}^{2} and its radius is 43\text{ mm}.

Round your answer to the nearest whole number.

5

A square pyramid has a surface area of 310\text{ cm}^{2}, and the area of its square base is 25\text{ cm}^2.

a

Find the area of each triangular face.

b

Find the height of each triangular face.

c

Find the perpendicular height of the pyramid correct to one decimal place.

6

A cone has a surface area of 560\text{ cm}^{2} and a radius of 6\text{ cm}.

a

Find the slant height of the cone. Round your answer to two decimal places.

b

Find the perpendicular height of the cone. Round your answer to one decimal place.

7

A sphere has a surface area of 450\text{ cm}^{2}. Find its radius correct to two decimal places.

8

A ball has a surface area of 50.27\text{ mm}^2. Find its radius to two decimal places.

9

A ball has a surface area of 1809.56\text{ m}^2.

a

Find its radius to the nearest metre.

b

Find the volume of the ball to two decimal places.

Work backwards from volume and capacity
10

The following rectangular prism has a volume of 168\text{ mm}^3:

Find the length, a, of the rectangular prism.

11

The following rectangular prism has a volume of 1680\text{ mm}^3:

Find the width of the rectangular prism in millimetres:

12

Find the side length of a cube with volume 27\text{ cm}^3.

13

Find the length of the rectangular prism with volume 162\text{ cm}^3, width 6\text{ cm} and height 3\text{ cm}.

14

The volume of the triangular prism shown is 231\text{ cm}^3.

Find the value of k.

15

The volume of the triangular prism shown is 287.5\text{ m}^{3}.

Find the value of k, to one decimal place.

16

The volume of the triangular prism shown is 247.45\text{ cm}^{3}.

Find the value of y to one decimal place.

17

A pentagonal prism has a volume of 176.4\text{ m}^3 and a cross sectional area of 29.4\text{ m}^2. Find the height of the prism.

18

Find the cross-sectional area of a pentagonal prism, if the volume is 470\text{ mm}^3 and the height is 10\text{ mm}.

19

A square pyramid has a base with side length 12\text{ mm} and a volume of 1248\text{ mm}^3. Find the height of the prism.

20

A rectangular pyramid has a volume of 432\text{ cm}^{3}. The height of the pyramid is 18\text{ cm} and the width of the base is 6\text{ cm}. Find the length of the base.

21

A rectangular pyramid has a volume of 288\text{ cm}^3. The base has a width of 12\text{ cm} and length 6\text{ cm}. Find the height of the pyramid.

22

A square pyramid has a height of 24\text{ cm} and a volume 2592\text{ cm}^3. Find the base side length of the pyramid.

23

A cone has a volume of 26\,640.97\text{ cm}^3 and radius 31.9\text{ cm}. Find the height of the cone. Round your answer to the nearest centimetre.

24

A cone has a volume of 1273.39\text{ mm}^{3} and height 19\text{ mm}. Find the radius of the cone to the nearest millimetre.

25

A cone has a volume of 196\text{ mm}^3. If the height and radius of the cone are equal in length, find the radius of the cone. Round your answer to two decimal places.

26

A sphere has a radius of r\text{ cm} and a volume of \dfrac{343 \pi}{3}\text{ cm}^3. Find the radius of the sphere to two decimal places.

27

If the volume of a sphere is 20\,579.526\text{ cm}^3, find the length of its diameter. Round your answer to one decimal place.

28

A hemispherical bowl has a capacity of 2.5\text{ L}. Find its radius in centimetres. Round your answer to one decimal place.

29

A cylindrical water tank has capacity 90\,000\text{ L} and a height of 2.5\text{ m}. Find the length of its diameter. Round your answer to one decimal place.

Applications
30

This nesting box needs to have a volume of 129\,978\text{ cm}^{3}, a height of 83\text{ cm} and a width of 54\text{ cm}.

Find the depth, d, of the box.

31

The volume of the following tent is 4.64\text{ m}^3.

Find the height, h, of the tent.

32

A swimming pool has the shape of a trapezoidal prism as shown in the diagram.

a

Find the volume of the pool.

b

If the pool is only three-quarters full, find the volume of the empty space in the pool.

c

If the distance from the water level to the top of the pool is h\text{ m} when it is is three-quarters full, find the value of h.

33

A cylindrical paddle-pool with radius 1.4\text{ m} has a water depth of 47\text{ cm}.

a

Find the volume of water in the pool, in cubic metres. Round your answer to three decimal places.

b

If 100\text{ L} of water is added, by how many centimetres does the depth of the water increase? Round your answer to one decimal place.

34

A spherical steel shot-put has a mass of 5.3\text{ kg}. Given that the density of the steel used is 8070\text{ kg/m}^3, find:

a

The volume of the shot-put in cubic centimetres. Round your answer to one decimal place.

b

The radius of the shot-put in centimetres. Round your answer to one decimal place.

35

Yuri has 6\text{ m}^2 of metal sheeting out of which he plans to make a raised garden bed. He is considering two designs:

  • Design 1: An open rectangular prism with height 0.4\text{ m}, and two sides 4.2\text{ m} long (no top or base), or

  • Design 2: An open cylinder with height 0.4\text{ m} (no top or base)

a

Find the length, l, of the other two sides of the rectangular design. Round your answer to two decimal places.

b

Find the radius, r, of the cylindrical design.

c

Which design will give the larger volume for the garden bed?

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Outcomes

ACMGM019

calculate the volumes of standard three-dimensional objects such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the volume of water contained in a swimming pool

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