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GCSE Maths Geometry and Measure Triangular Prism

Surface Area of a Triangular Prism

Surface Area of a Triangular Prism

Here we will learn about the surface area of a triangular prism and how to calculate it.

There are also volume and surface areaof a triangular prism 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 the surface area of a triangular prism?

The surface area of a triangular prism is the total area of all of the faces. 

To work out the surface area of a triangular prism, we need to work out the area of each face and add them all together.

Lateral faces are all of the faces of an object excluding the top and the base. For a triangular prism the top and the base are triangles and the lateral faces are rectangular sides. The lateral surface area of a triangular prism  is the total area of the rectangular sides

The triangular faces of a triangular prism are congruent (exactly the same) but, unless the triangle is isosceles or equilateral, the rectangles are all different.

E.g.

Since it is an area, surface area is measured in square units (e.g. mm^2, cm^2, m^2 etc).

What is the surface area of a triangular prism?

What is the surface area of a triangular prism?

How to calculate the surface area of a triangular prism

In order to work out the surface area of a triangular prism:

  1. Work out the area of each face.
  2. Add the five areas together.
  3. Include the units.

How to calculate the surface area of a triangular prism

How to calculate the surface area of a triangular prism

Surface area of a triangular prism worksheet

Get your free Surface area of a triangular prism worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Surface area of a triangular prism worksheet

Get your free Surface area of a triangular prism worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Surface area of a triangular prism examples

Example 1: finding the surface area of a triangular prism with a right triangle

Work out the surface area of the triangular prism

  1. Work out the area of each face.

The area of the triangle at the front is \frac{1}{2}\times12\times5 = 30cm^{2}

The back face is the same as the front face so the area of the back is also 30cm^{2} .

The area of the base is 9\times12=108cm^{2}

The area of the left side is 9\times5=45cm^{2}

The area of the top is 9\times13=117cm^{2}

It will make our working clearer if we use a table:

FaceArea
Front½ × 12 × 12 × 5 = 30
Back30
Bottom9 × 12 = 108
Left side9 × 5 = 45
Top9 × 13 = 117

2Add the five areas together.

Total surface area: 30 + 30 + 108 + 45 + 117 = 330

3Include the units.

The measurements on the triangular prism are in cm therefore the total surface area of the triangular prism = 330cm^{2} .

Example 2: surface area of a triangular prism with an isosceles triangle

Work out the surface area of the triangular prism

FaceArea
Front½ × 6 × 9.5 = 28.5
Back28.5
Bottom6 × 14 = 84
Left side10 × 14 = 140
Right side10 × 14 = 140

Total surface area: 28.5 + 28.5 + 84 + 140 + 140 = 421

The measurements on the triangular prism are in mm therefore the total surface area of the triangular prism = 421mm^{2} .

Example 3: surface area of a right triangular prism

This prism has a triangular base and rectangular sides. We are told the height of the prism is 9cm . We can work out the surface area in exactly the same way, we just adjust the labels we give to each face in our table.

FaceArea
Top½ × 6 × 8 = 24
Bottom24
Front9 × 10 = 90
Left side9 × 6 = 54
Right side9 × 8 = 72

Total surface area: 24 + 24 + 90 + 54 + 72 = 264

The measurements on the triangular prism are in cm therefore the total surface area of the triangular prism = 264cm^{2} .

Example 4: surface area of a triangular prism

Work out the surface area of the triangular prism

FaceArea
Front½ × 35 × 12 = 210
Back210
Bottom40 × 35 = 1400
Left side40 × 12 = 480
Top40 × 37 = 1480

Total surface area: 210 + 210 + 1400 + 480 + 1480 = 3780

The measurements on the triangular prism are in m therefore the total surface area of the triangular prism = 3780m^{2} .

Example 5: surface area of a triangular prism with different units

Work out the surface area of the triangular prism. Give your answer in mm^{2} .

Some of the measurements here are in cm and some are in mm. Since we have been asked for the answer in mm^{2} , we need to convert all measurements to mm :

1.4cm = 14mm and 2.5cm=25mm .

FaceArea
Front½ × 8 × 14 = 56
Back56
Bottom25 × 8 = 200
Left side25 × 14 = 350
Right side25 × 16.1 = 402.5

Total surface area: 56 + 56 + 200 + 350 + 402.5 = 1064.5

The measurements we have used are in mm therefore the total surface area of the triangular prism = 1064.5mm^{2} .

Example 6: surface area when there is a missing length

Work out the surface area of the triangular prism. Give your answer in cm^{2} .

When calculating the surface area we need the length of each side of the triangle.

We are told the base and height of the triangle and the length of the prism but we don’t have the length of the hypotenuse of the triangle.

To work this out we can use the Pythagorean Theorem a^{2}+b^{2}=c^{2} .

\[ 8^{2}+15^{2} = c^{2} \\ 289 = c^{2} \\ \sqrt{289} = c \\ 17 = c \]

FaceArea
Front½ × 8 × 15 = 60
Back60
Bottom9 × 8 = 72
Left side9 × 15 = 135
Right side9 × 17 = 153

Total surface area: 60 + 60 + 72 + 135 + 153 = 480

The measurements we have used are in cm therefore the total surface area of the triangular prism = 480cm^{2} .

Common misconceptions

  • 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.

  • Thinking all of the rectangles have the same area

Usually all of the rectangle have different areas (unless the triangle is isosceles or equilateral).

  • Using the wrong measurements to work out the area of the triangle faces

In surface area questions, we need to know all three side lengths of the triangle however we only need the base and the height to calculate the area of the triangle

Surface area of a triangular prism is part of our series of lessons to support revision on triangular prism. You may find it helpful to start with the main triangular prism 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:

Practice surface area of a triangular prism questions

1. Work out the surface area of the triangular prism
 

 

1200 \mathrm{cm}^{2}

GCSE Quiz False

1040 \mathrm{cm}^{2}

GCSE Quiz False

920 \mathrm{cm}^{2}

GCSE Quiz True

1140 \mathrm{cm}^{2}

GCSE Quiz False

Work out the surface area of each face:

 

Face Area
Front ½ × 8 × 15 = 60
Back 60
Bottom 20 × 8 = 160
Left side 20 × 15 = 300
Right side 20 × 17 = 340

 
\text{Total surface area }=60+60+160+300+340=920\mathrm{cm}^{2}

2. Work out the surface area of the triangular prism:
 

 

720 \mathrm{cm}^{2}

GCSE Quiz False

672 \mathrm{cm}^{2}

GCSE Quiz False

546 \mathrm{cm}^{2}

GCSE Quiz False

576 \mathrm{cm}^{2}

GCSE Quiz True

Work out the surface area of each face:

 

Face Area
Front ½ × 12 × 8 = 48
Back 48
Bottom 15 × 12 = 180
Left side 15 × 10 = 150
Right side 15 × 10 = 150

 
\text{Total surface area }=48+48+180+150+150=576\mathrm{cm}^{2}

3. Work out the surface area of the triangular prism:
 

 

82 \mathrm{cm}^{2}

GCSE Quiz True

94 \mathrm{cm}^{2}

GCSE Quiz False

36 \mathrm{cm}^{2}

GCSE Quiz False

66 \mathrm{cm}^{2}

GCSE Quiz False

Work out the surface area of each face:

 

Face Area
Front ½ × 3 × 4 = 6
Back 6
Top 6 × 3 = 18
Left side 6 × 4 = 24
Right side 6 × 5 = 30

 
\text{Total surface area }=6+6+18+24+30=84\mathrm{cm}^{2}

4. Work out the surface area of the triangular prism
 

 

7.007 \mathrm{cm}^{2}

GCSE Quiz False

14.014 \mathrm{cm}^{2}

GCSE Quiz False

24.04 \mathrm{cm}^{2}

GCSE Quiz True

30.89 \mathrm{cm}^{2}

GCSE Quiz False

Work out the surface area of each face:

 

Face Area
Front ½ × 2.6 × 4.9 = 6.37
Back 6.37
Bottom 1.1 × 2.6 = 2.86
Left side 1.1 × 4.9 = 5.39
Right side 1.1 × 5.5 = 6.05

 
\text{Total surface area }=6.37+6.37+2.86+5.39+6.05=27.04\mathrm{cm}^{2}

5. Work out the surface area of the triangular prism. Give your answer in square centimetres.
 

 

15200 \mathrm{cm}^{2}

GCSE Quiz True

14400 \mathrm{cm}^{2}

GCSE Quiz False

2448.8 \mathrm{cm}^{2}

GCSE Quiz False

96000 \mathrm{cm}^{2}

GCSE Quiz False

Notice that some of the measurements are in m and some are in cm . Since we are asked to give the answer in square centimetres, we need to convert all the measurements to cm .
 
0.5m = 50cm and 0.8m=80cm .

 

Next, Work out the surface area of each face:

 

Face Area
Front ½ × 60 × 40 = 1200
Back 1200
Bottom 80 × 60 = 4800
Left side 80 × 50 = 4000
Right side 80 × 50 = 4000

 
\text{Total surface area }=1200+1200+4800+4000+4000=15200 \mathrm{cm}^{2}

6. Work out the surface area of the triangular prism

330 \mathrm{m}^{2}

GCSE Quiz False

936 \mathrm{m}^{2}

GCSE Quiz False

1560 \mathrm{m}^{2}

GCSE Quiz False

660 \mathrm{m}^{2}

GCSE Quiz True

In this question, we are missing the height of the triangle. Since it is a right angled triangle, we can use Pythagoras’ theorem to work out the height:

 

 

\begin{aligned} a^{2}+b^{2}&=c^{2}\\ h^{2}+12^{2}&=13^{2}\\ h^{2}+144&=169\\ h^{2}&=169-144\\ h^{2}&=25\\ h&=5 \mathrm{m} \end{aligned}

 

Next, work out the surface area of each face:
 

Face Area
Front ½ × 12 × 5 = 30
Back 30
Bottom 20 × 12 = 240
Top 20 × 13 = 260
Left side 20 × 5 = 100

 
\text{Total surface area }=30+30+240+260+100=660 \mathrm{m}^{2}

Surface area of a triangular prism GCSE questions

1. Work out the surface area of the triangular prism.
 

(3 marks)

Show answer

\frac{1}{2} \times 0.3 \times 0.4 = 0.06

(1)

1 \times 0.3=0.1, ~1 \times 0.4 – 0.4,~ 1 \times 0.5 = 0.5

(1)

0.06+0.06+0.3+0.3+0.5=1.32 \mathrm{m}^{2}

(1)

2. A packaging company wants to minimise the amount of packaging they use. Which of these shapes should they choose to make their packaging? Show how you decide.
 

 

(5 marks)

Show answer

 
Prism A: Surface area = 24+24+90+120+150

(1)

 
\text{Surface area }=408 \mathrm{cm}^{2}

(1)

 
Prism B: Surface area = 25+25+140+70+156.8

(1)

 
\text{Surface area }=416.8 \mathrm{cm}^{2}

(1)

 
They should use shape A

(1)

3. A packaging box is made in the following shape
 

The material used to make the box costs 0.15p per \mathrm{cm}^{2} to produce. How much would it cost to make 100 boxes? Give your answer in pounds.
 
 

(5 marks)

Show answer

 
Surface area = 12+12+88+55+55

(1)

 
\text{Surface area }=222 \mathrm{cm}^{2}

(1)

 
222 \times 0.15 = 33.3

(1)

 
33.3 \times 100=3330 \text{p}

(1)

 
3330p=£33.30

(1)

Learning checklist

You have now learned how to:

  • Calculate the surface area of a triangular prism
  • Use the properties of faces, surfaces, edges and vertices of cubes and cuboids to solve problems in 3-D

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