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5.052 Pyramids and cones

Worksheet
Surface area of pyramids
1

A square pyramid has a base area of 20\text{ cm}^2 and each triangular face has an area of 15\text{ cm}^2.

Find the surface area of the pyramid.

2

A regular hexagonal pyramid has a base area of 36\text{ cm}^2 and each triangular face has an area of 14\text{ cm}^2.

Find the surface area of the pyramid.

3

Find the surface area of the following pyramids:

a
b
4

Find the surface area of the following pyramids:

a

A square pyramid with base edge 10\text{ cm} and perpendicular height 5\text{ cm}.

b

A rectangular pyramid with base 10\text{ cm} by 5\text{ cm}, and perpendicular height 4\text{ cm}.

5

Consider the following square pyramid:

a

Find the height of the triangular faces, correct to two decimal places.

b

Find the surface area of the square pyramid correct to one decimal place.

6

Consider the rectangular pyramid with rectangular base of dimensions 4\text{ cm} by 8\text{ cm} and perpendicular height of 8\text{ cm}.

a

Find the value of x, correct to two decimal places.

b

Find the value of y, correct to two decimal places.

c

Find the surface area of the pyramid, correct to one decimal place.

Surface area of cones
7

Find the surface area of the following cones, correct to two decimal places:

a
b
8

For each of the following cones, use their given measurements to:

i

Find the slant height of the cone.

ii

Calculate the surface area of the cone, correct to two decimal places.

a

Radius =3\text{ cm}, Height =4\text{ cm}

b

Radius =5\text{ cm}, Height =12\text{ cm}

c

Radius =12\text{ cm}, Height =5\text{ cm}

d

Radius =15\text{ cm}, Height =8\text{ cm}

e

Radius =1\text{ m}, Height =1\text{ m}

f

Radius =8\text{ m}, Height =5\text{ m}

9

A cone has a radius of 8\text{ cm} and a height of 6\text{ cm}. Find the exact surface area of the cone. Give your answer in terms of \pi.

10

A cone has a diameter of 8\text{ units} and a height of 7\text{ units}. Find the surface area of the cone, correct to two decimal places.

Volume of pyramids
11

Find the volume of the following pyramids using the given measurements.

a

Base area =10\text{ cm}^2, Height =9\text{ cm}

b

Base area =15\text{ m}^2, Height =8\text{ m}

c

Base area =16\text{ cm}^2, Height =12\text{ cm}

d

Base area =130\text{ mm}^2, Height =7\text{ mm}

12

Find the volume of the following pyramids, rounding your answers to two decimal places where necessary:

a
b
c
d
e
f
13

The Great Pyramid in Egypt has a square base with side length 230 \text{ m} and a vertical height of 146 \text{ m}. Find the volume of the Great Pyramid.

Volume of cones
14

Find the volume of a cone that has radius of 8\text{ cm} and vertical height of 13\text{ cm}, correct to two decimal places.

15

Find the exact volumes of the following cones using their given measurements.

a

Radius =3\text{ cm}, Height =8\text{ cm}

b

Radius =4\text{ cm}, Height =6\text{ cm}

c

Radius =10\text{ cm}, Height =3\text{ cm}

d

Radius =12\text{ cm}, Height =15\text{ cm}

e

Radius =5\text{ cm}, Height =4\text{ cm}

f

Radius =10.8\text{ cm}, Height =7\text{ cm}

16

Find the volume of a cone that has a diameter of 3\text{ m} and a height of 1.3\text{ m}, correct to two decimal places.

17

Consider the cone on the right:

a

Calculate the perpendicular height.

b

Hence find the volume of the cone. Round your answer to three significant figures.

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Outcomes

1.3.3

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

1.3.4

calculate the surface areas of standard three-dimensional objects, such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations; for example, the surface area of a cylindrical food container

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