Text
Unit C3 Section 4
Surface Area

The net of a cube can be used to find its surface area.

The net is made up of 6 squares, so the surface area will be 6 times the area of one square. If x is the length of the sides of the cube its surface area will be 6x2.

This diagram shows the net for a cuboid. To find the surface area the area of each of the 6 rectangles must be found and then added to give the total.

If x, y and z are the lengths of the sides of the cuboid, then the area of the rectangles in the net are as shown here.

The total surface area of the cuboid is then given by

A = 2xy + 2xz + 2yz

To find the surface area of a cylinder, consider how a cylinder can be broken up into three parts, the top, bottom and curved surface.

The areas of the top and bottom are the same and each is given by πr2.

The curved surface is a rectangle. The length of one side is the same as the circumference of the circles, 2πr, and the other side is simply the height of the cylinder, h. So the area is 2πrh.

The total surface area of the cylinder is

2πr2 + 2πrh

Another important result is the surface area of a sphere.

For a sphere with radius r, the surface area is given by the formula

4πr2

Worked Examples

1

Find the surface area of the cuboid shown in the diagram.

The diagram shows the net of the cuboid and the areas of the rectangles that it contains.

Using the net, the total surface area is given by

A = 2 × 20 + 2 × 30 + 2 × 24
= 148 cm²
2

Cans are made out of aluminium sheets, and are cylinders of radius 3 cm and height 10 cm. Find the area of aluminium needed to make one can.

The diagram shows the two circles and the rectangle from which cans will be made.

The rectangle has one side as 10 cm, the height of the cylinder and the other side is 2 × π × 3 cm, the circumference of the top and bottom.

The area of the rectangle is 10 × 2 × π × 3 The area of each circle is π × 32

So the total surface area is A = 10 × 2 × π × 3 + 2 × π × 32
= 245.04 cm² (to 2 d.p.)
3

A ball has radius 4 cm. What is its surface area, to the nearest cm²?

Surface area = 4πr2 cm²
= 4π42 cm²
= 64π cm²
= 201 cm² to the nearest cm²

Note

There is a formula for calculating the surface area of a cone:

surface area of cone = πrs + πr2

where

s = slant height of the cone

r = radius of the base

(and s2 = h2 + r2, where h is the perpendicular height of the cone).

4

What is the surface area of a cone of base radius 5 cm and perpendicular height 12 cm?
Give your answer in terms of π.

Slant height =
=
=
= 13 cm
Surface area = (π × 5 × 13 + π × 52) cm2
= (65π + 25π) cm2
= 90π cm2

Note

There is also a formula for calculating the surface area of a square-based pyramid:

surface area = 2as + a2

where

s = perpendicular slant height of the pyramid

a = length of the side of the square base

(and h is the perpendicular height of the pyramid).

Note that a2 is the surface area of the base and each trianglar face has area as.

5

What is the surface area of a square-based pyramid of base side 6 cm and height 4 cm?

We first calculate the slant height from

s2 = h2 + 32
= 42 + 32
= 16 + 9
s2 = 25 ⇒ s = 5 cm

Hence,

surface area = (2 × 6 × 5 + 62) cm2
= (60 + 36) cm2
= 96 cm2

Exercises

Find the surface area of each of the following cubes or cuboids.

(a)
cm²
(b)
cm²
(c)
cm²
(d)
cm²
(e)
cm²
(f)
m²

Find the total surface area of each cylinder shown below.

(a)
cm² (to 2 d.p.)
(b)
cm² (to the nearest cm²)
(c)
cm² (to the nearest cm²)
(d)
m² (to the nearest m²)
(e)
m² (to 1 d.p.)
(f)
cm² (to the nearest cm²)

A matchbox consists of a tray that slides into a sleeve. If the tray and sleeve have the same dimensions and no material is used up in joins, find:

(a)

the area of cardboard needed to make the tray,

cm²
(b)

the area of cardboard needed to make the sleeve,

cm²
(c)

the total area of the cardboard needed to make the matchbox.

cm²

A car tyre can be thought of as a hollow cylinder with a hole cut out of the centre. Find the surface area of the exterior surfaces of the tyre (to the nearest whole cm²).

cm²

The diagram shows a cuboid.

The co-ordinates of P are (3, 4, 0).

The co-ordinates of Q are (3, 9, 0).

The co-ordinates of C are (–1, 9, 6).

(a)

Write down the (x, y, z) co-ordinates

(i)
of R (, , )
(ii)
of B (, , )
(b)

Write down the lengths of each of the following edges of the cuboid.

(i)
PQ units
(ii)
QR units
(c)

Calculate the total surface area of the cuboid.

square units

A beach ball has radius 30 cm. What is its surface area, to the nearest tenth of m2?

m²

What is the total surface area of a square-based cone of perpendicular height 8 cm and side length 5 cm? Give your answer to 1 decimal place.

cm²

Information

Smaller animals have more surface area compared to their volume than larger animals. Because of this, smaller animals tend to lose water and body heat more easily than larger animals. Children have 2 times as much surface area compared to volume as adults. Thus children are more prone to dehydration and hypothermia.