# How To Calculate Surface Area

Here we will learn how to calculate the surface area of a variety of three-dimensional shapes, including cuboids, prisms, cylinders, cones and spheres.

There are also surface area worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is surface area?

The surface area of a three dimensional shape is the total area of all of the faces.

To find the surface area of a shape, we find the area of each face and add them together.

E.g.

Total surface area

\begin{aligned} &= 6+6+24+30+18 \\ &=84cm^2 \end{aligned}

The measurements here are in centimetres so the surface area is measured in square centimetres (cm^2) .

## How to calculate surface area

In order to calculate surface area:

1. Calculate the area of each face.
3. Write the answer, including the units.

### Related lessons on 3D shapes

How to calculate surface area is part of our series of lessons to support revision on 3D shapes. You may find it helpful to start with the main 3D shapes lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

## Surface area examples

### Example 1: surface area of a cuboid

Work out the surface area of the cuboid. We can think of this as finding the surface area of a rectangular prism or the  surface area of a box.

1. Calculate the area of each face.

The sum of the areas is: 40+40+60+60+24+24=248

3Write the answer, including the units.

The measurements are in cm so the surface area will be measured in cm^2 .

Total surface area = 248cm^2

### Example 2: surface area of a cube

Work out the surface area of the cube:

Calculate the area of each face.

Write the answer, including the units.

### Example 3: surface area of a triangular prism

Work out the surface area of the triangular prism

Calculate the area of each face.

Write the answer, including the units.

### Example 4: surface area of a cylinder

The radius of this cylinder is 5m and the height of the cylinder is 2m. Work out the surface area of the cylinder. Give your answer to 3 significant figures.

Calculate the area of each face.

Write the answer, including the units.

### Example 5: surface area of a cone

Work out the surface area of the cone. Give your answer to 1 decimal place.

Calculate the area of each face.

Write the answer, including the units.

### Example 6: surface area of a sphere

Work out the surface area of the sphere. Give your answer to the nearest integer.

Calculate the area of each face.

Write the answer, including the units.

### Common misconceptions

• Calculating with different units

You need to make sure all measurements are in the same units before calculating surface area. (For example you can’t have some measurements in cm and some in m ).

• Make sure you have the correct units

For area we use square units such as cm^2 (square centimetres), m^2 (square metres), in^2 (square inches) and ft^2 (square feet),

For volume we use cube units such as cm3, \; m^3, \; km^3

• Calculating volume instead of surface area

Volume and surface area are different things – volume tells us the space within the shape whereas surface area is the total area of the faces. To find surface area, work out the area of each face and add them together.

• Rounding

It is important to not round decimals until the end of the calculation. Rounding too early will result in an inaccurate answer. Keep as many values in terms of \pi until you need to calculate the final rounded answer.

### Practice surface area questions

1. Find the surface area of the rectangular prism below.

92cm^2

48cm^2

108cm^2

76cm^2

Calculating the area of each face, we have:

Total surface area: 16+16+24+24+6+6=92cm^2

2. Calculate the surface area of the triangular prism.

360cm^2

240cm^2

300cm^2

480cm^2

Calculating the area of each face, we have:

Total surface area = 30+30+96+104+40=300cm^2

3. Work out the surface area of the prism.

464mm^2

432mm^2

400mm^2

440mm^2

Calculating the area of each face, we have:

Total surface area = 20+20+160+40+100+100=440mm^2

4. Find the surface area of the cylinder. Give your answer to 3 significant figures.

1190m^2

633m^2

449m^2

346m^2

\begin{aligned} \text{Curved surface area }&=2 \pi rh\\\\ &=2 \times \pi \times 6.5 \times 9\\\\ &=117\pi \end{aligned}

\begin{aligned} \text{Area of a circle }&=\pi r^{2}\\\\ &=\pi \times 6.5^{2}\\\\ &=\frac{169}{4}\pi \end{aligned}

Total surface area: 117\pi+\frac{169}{4}\pi+\frac{169}{4}\pi=\frac{403}{2}\pi

\frac{403}{2}\pi=633.0309197…

Surface area = 633m^2 \; (3sf)

5. Work out the surface area of the cone. Give your answer to 3 decimal place.

29.5cm^2

22.6cm^2

7.5cm^2

23.9cm^2

\begin{aligned} \text{Curved surface area }&=\pi rl\\\\ &=\pi \times 2 \times 2.7\\\\ &=5.4\pi \end{aligned}

\begin{aligned} \text{Area of a circle }&=\pi r^{2}\\\\ &=\pi \times 2^{2}\\\\ &=4\pi \end{aligned}

Total surface area: =5.4\pi+4\pi=9.4\pi

Surface area = 29.5cm^2 \; (1dp)

6. Calculate the surface area of the sphere. Give your answer to 2 decimal place.

201.06m^2

2.01m^2

0.27m^2

25.27m^2

\begin{aligned} \text{Surface area of a sphere }&=4 \pi r^{2}\\\\ &=4 \times \pi \times 0.4^{2}\\\\ &=0.64\pi \\\\ &=2.010619298… \end{aligned}

Surface area = 2.01m^2 \; (2dp)

### Surface area GCSE questions

1. Calculate the surface area of the cone. Give your answer to the nearest integer.

(4 marks)

\begin{aligned} \text{Curved surface area }&=\pi \times 21 \times 37\\\\ &=2441.017492…\text{ or }777\pi \end{aligned}

(1)

\begin{aligned} \text{Area of circle }&=\pi \times 21^{2}\\\\ &=1385.4423…\text{ or }441\pi \end{aligned}

(1)

\text{Total surface area } = \text{their }777\pi+ \text{their } 441\pi =1218\pi =3826.459852…

(1)

\text{Surface area } = 3826cm^2

(1)

2. A drinks company is designing a new product. Can A has a volume of 108\pi \text{ cm}^3 and Can B has a volume of 98\pi \text{ cm}^3.

The company wants to minimise their packaging for the new product. Which should they choose? Explain your answer.

(9 marks)

Can A

\begin{aligned} \text{Curved surface area }&=2 \times \pi\times 3 \times 12\\\\ &=226.1946711…\text{ or }72\pi \end{aligned}

(1)

\begin{aligned} \text{Area of circle }&=\pi \times 3^{2}\\\\ &=28.27433388…\text{ or }9\pi \end{aligned}

(1)

\text{Total surface area: }\text{their }72\pi+(2\times\text{their }9\pi)=90\pi=282.7433388…

(1)

Can B :
\begin{aligned} \text{Curved surface area }&=2 \times \pi\times 3.5 \times 8\\\\ &=175.9291886…\text{ or }56\pi \end{aligned}

(1)

\begin{aligned} \text{Area of circle }&=\pi \times 3.5^{2}\\\\ &=38.48451001…\text{ or }12.25\pi \end{aligned}

(1)

\text{Total surface area: }\text{their }56\pi+(2\times\text{their }12.25\pi)=80.5\pi=252.8982086…

(1)

Volume to Surface Area ratio for Can A

\begin{aligned} &108 \pi :90 \pi \\\\ &=1:0.83 \end{aligned}

(1)

Volume to Surface Area for Can B

\begin{aligned} &98 \pi :80.5 \pi \\\\ &=1:0.82 \end{aligned}

(1)

They should use can B because it has a smaller volume to surface area ratio.

(1)

3. The surface area of this sphere is 200cm^2. Calculate the radius of the sphere. Give your answer to 3sf.

(4 marks)

4\pi r^{2}=200

(1)

r^{2}=\frac{200}{4\pi}=15.91549431…

(1)

r=\sqrt{15.91549431…}=3.989422804…

(1)

r=4.00cm \; (3sf)

(1)

## Learning checklist

You have now learned how to:

• Find the surface area of cuboids and prisms
• Find the surface area of cylinders, cones and spheres
• Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3D

## Still stuck?

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