Unit 9 Section 4 : Surface Area and Volume of 3-D Shapes

In this section we calculate the volume and surface area of 3-D shapes such as cubes, cuboids, prisms and cylinders.

Cube Volume = x³
Surface area = 6x²
Cuboid Volume = xyz
Surface area = 2xy + 2xz + 2yz
Cylinder Volume = π r²h
Area of curved surface = 2π rh
Area of each end = π r²
Total surface area = 2π rh + 2π r²
Prism A prism has a uniform cross-section
Volume = area of cross section × length = A l

Example 1

(a)

Calculate the volume of the cuboid shown.

Volume = 4 × 18 × 5 = 360 m³
(b)

Calculate the surface area of the cuboid shown.

Surface area = (2 × 4 × 18) + (2 × 4 × 5) + (2 × 5 × 18)
= 144 + 40 + 180
= 364 m²

Example 2

Calculate the volume and total surface area of the cylinder shown.

Volume  =  π r²h  =  π × 4² × 6 = 96 π
 =  301.5928947 cm³
 =  302 cm³ (to 3 significant figures)
Area of curved surface  =  2π rh  =  2 × π × 4 × 6
 =  48π
 =  150.7964474 cm²
Area of each end  =  π r²  =  π × 4²
 =  16π
 =  50.26548246 cm²
Total surface area  =  150.7964474 + (2 × 50.26548246)
 =  251.3274123 cm²
 =  251 cm² (to 3 significant figures)

Note: From the working we can see that the area of the curved surface is 48π, and that the area of each end is 16π. The total surface area is therefore

48π + (2 × 16π) = 80π = 251.3274123 cm²
= 251 cm² (to 3 significant figures)

Example 3

Calculate the volume of this prism.

Area of end of prism  =  × 8 × 6
 =  24 cm²
Volume of prism  =  24 × 10
 =  240 cm³

Exercises

Question 1

Calculate the volume and surface area of each of the following cuboids:

(a)

Volume = cm³

Surface area = cm²

(b)

Volume = m³

Surface area = m²

Question 2

Giving your answers correct to 3 significant figures, calculate the volume and total surface area of each of the following cylinders:

(a)

Volume = cm³

Surface area = cm²

(b)

Volume = cm³

Surface area = cm²

Question 3

Calculate the volume of each of the following prisms:

(a)

cm³

(b)

m³

Question 4

Calculate the volume and surface area of the following prism:

Volume = m³

Surface area = m²

Question 5

The diagram shows a wooden block that has had a hole drilled in it. The diameter of the hole is 2 cm.
Calculate the volume of this solid, giving your answer correct to 2 decimal places.

cm³
Volume = block – hole = 4 × 6 × 6 – 1² × π × 6 = 144 – 6π = 125.15 (to 2 d.p.)
Question 6

A concrete beam is to rest on two concrete pillars. The beam is a cuboid with sides of length 0.5m, 3m and 0.4m. The pillars have diameter 0.4m and height 2m. Calculate the total volume of concrete needed to make the beam and the pillars. Round your answer to a sensible level of accuracy.

m³
Question 7

The diagram shows the cross-section of a pipe of length 50 cm.
The inner diameter of the pipe is 20 cm and the outer diameter is 30 cm.

(a)

Calculate the volume of metal needed to make the pipe. Round your answer to a sensible level of accuracy.

cm³
(b)

Calculate the total surface area of the pipe, including the inside surface.
Round your answer to 3 significant figures.

cm²
Total surface area = 2 × (15² – 10²) × π + 30π × 50 + 20π × 50
= 250π + 1500π + 1000π = 2750π
= 8639.379797 cm² = 8640 cm² (to 3 s.f.)
Question 8

The diagram shows a prism.
The cross-section of the prism consists of a rectangle and a semicircle.

(a)

Calculate the volume of the prism. Give your answer to the nearest cm³.

cm³
(b)

Calculate the total surface area of the prism. Give your answer to the nearest cm².

cm²
Question 9

The volume of the prism shown is 720 mm³.

(a)

Determine the length of the prism.

mm
(b)

Calculate the surface area of the prism.

mm²
Question 10

A cylinder has a diameter of 12 cm and a curved surface area of 132π or 415 cm² (to 3 significant figures).

(a)

Determine the height of the cylinder.

cm
(b)

Calculate the volume of the cylinder, giving your answer to the nearest cm³.

cm³
Question 11

(a)

These cuboids are made from small cubes. Write how many small cubes there are in each cuboid. The first is done for you.

(i) 12 small cubes
(ii) small cubes
(iii) small cubes
(iv) small cubes
(b)

This shape is made with two cuboids. Write how many cubes there are in this shape.

cubes
Question 12

(a)

What is the volume of this standard size box of salt?

cm³
(b)

What is the volume of this special offer box of salt, which is 20% bigger?

cm³

The standard size box contains enough salt to fill up 10 salt pots.

(c)

How many salt pots may be filled up from the special offer box of salt?

pots
Question 13

(a)

Look at this triangle.
Is x a right angle?

by theorem
(b)

What is the volume of this prism?

cm³

Area of cross-section = × base × height = × 6 × 8 = 24 cm²

Volume of prism = area of cross-section × length = 24 × 7 = 168cm³

(c)

Prisms A and B have the same cross-sectional area.

Complete the table:

Prism APrism B
height5 cm3 cm
volume200 cm³ cm³
Question 14

TJ's Cat Food is sold in tins shaped like this.
Each tin has an internal height of 5 cm.

(a)

The area of the lid of the tin is 35 cm². Work out the volume of cat food that the tin contains.

cm³
(b)

The label that goes round the tin overlaps by 1 cm.

The area of the label is 134 cm². Work out the distance around the tin.

cm

Length of label = 134 ÷ 5 = 26.8 cm

Distance around tin = 26.8 – 1 = 25.8 cm

TJ's Cat Food plans to use tins that are the shape of cylinders. The internal measurements of a tin are shown.

(c)

Work out the volume of cat food that the tin contains.

cm³ (to 3 significant figures)
Volume = πr²h = π × 3² × 4 = 36π = 113.0973355 cm³
= 113 cm³ (to 3 s.f.)