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

There are also volume and surface area of a prism worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβre still stuck.

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

To work out the surface area of a prism, you work out the area of each face and add them all together.

Sketching each face of the prism and labelling its dimensions can help to structure the solution. Below are a few examples of prisms with their component faces.

If we inspect a cuboid, we can see that the prism is made up of 6 rectangular faces.

It specifically has 3 pairs of congruent faces (not all prisms have this property) as the opposing faces are the same size.

To calculate the surface area of the cuboid, we need to calculate the area of each face, and then add them together.

Now that we know the area of each face, the surface area of the prism is the sum of these values.

40+24+15+40+24+15=158.The surface area of the cuboid is equal to 158cm^2 .

**Note: **Surface area is measured in square units (e.c. mm^2, cm^2, m^2 etc).

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

**Work out the area of each face.****Add the area of each face together.****Write the answer, including the units.**

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

COMING SOONGet your free surface area of a prism worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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

Work out the surface area of the prism.

**Work out the area of each face.**

The area of the front of the prism is \frac{1}{2} \times 4 \times 3=6\mathrm{cm}^{2} . The back face is the same as the front face so the area of the back face is also 6cm^2 .

The area of the base is 4\times2=8cm^2 .

The area of the left side is 2\times3=6cm^2 .

The area of the right side is 2\times5=10cm^2 .

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

Face | Area |
---|---|

Front | \frac{1}{2} \times 4 \times 3 = 6 |

Back | 6 |

Bottom | 4\times 2 = 8 |

Left side | 2\times 3 = 6 |

Top | 2\times 5 = 10 |

2**Add the area of each face together.**

The total surface area (SA) is SA=6+6+8+6+10=36 .

3**Write the answer, including the units.**

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

Surface area = 36cm^2 .

Work out the surface area of the prism.

**Work out the area of each face.**

A cuboid has 6 faces, with 3 pairs of identical faces.

Face | Area |
---|---|

Front | 7\times 2 = 14 |

Back | 14 |

Bottom | 7\times 3 = 21 |

Top | 21 |

Left side | 2\times 3 = 6 |

Right side | 6 |

**Add the area of each face together.**

The total surface area (SA) is SA=14+14+21+21+6+6=82 .

**Write the answer, including the units.**

The measurements on this prism are in m so the surface area of the prism is 82m^2 .

Work out the surface area of the prism.

**Work out the area of each face.**

A trapezoidal prism has 6 faces, with identical trapeziums at either end. Notice that all of the other faces are rectangular.

Face | Area |
---|---|

Front | \frac{1}{2}(8+18)\times{12}=156 |

Back | 156 |

Bottom | 20\times 18 = 360 |

Top | 20\times 8 = 160 |

Left side | 20\times 3 = 260 |

Right side | 260 |

**Add the area of each face together.**

The total surface area (SA) is SA=156+156+360+160+260+260=1352 .

**Write the answer, including the units.**

The measurements for this prism are in cm.

The surface area of the prism is 1352cm^2 .

Work out the surface area of the prism.

**Work out the area of each face.**

A parallelogram prism has 6 faces and, like a cuboid, it has 3 pairs of identical faces. Notice that, once again, all of the other faces are rectangular.

In this example some of the measurements are in cm and some are in m so before we begin we must convert the units so that they are the same. Here, we will convert all the units to metres ( m ), 40cm = 0.4m and 50cm = 0.5m .

Face | Area |
---|---|

Front | 1.2\times 0.4 = 0.48 |

Back | 0.48 |

Bottom | 1.2\times 1.5 = 1.8 |

Top | 1.8 |

Left side | 0.5\times 1.5 = 0.75 |

Right side | 0.75 |

**Add the area of each face together.**

The total surface area (SA) is SA=0.48+0.48+1.8+1.8+0.75+0.75=6.06 .

**Write the answer, including the units.**

The measurements that we have used are in m so the surface area of the prism is 6.06m^2 .

The surface area of a prism is special because all of the lateral faces are rectangles. To calculate the surface area of a prism, we calculate twice the area of the cross section and add the perimeter of the cross section multiplied by the depth of the prism.

Here we have a cuboid with a width of 4cm , a height of 1cm , and a depth of 2cm .

We will take the front and back faces of the cuboid as the cross section. The area of the cross section is equal to 4 x 1 = 4cm^2 .

As the front and back are congruent, the area of these two faces is equal to 4 x 2=8cm^2 .

Now, if we highlight the perimeter of the cross section, then unfold the cuboid so we can see the net, we can see that the perimeter of the cross section is the same as the height of the net.

As the lateral faces are all rectangles, the area of the lateral faces is equal to the perimeter of the cross section, multiplied by the depth of the prism.

(4+1+4+1) Γ 2 = 20cm^2Adding the area of the front and back to this value, we get 20+8=28 .

The surface area of the cuboid is 28cm^2 .

In general, the surface area of any prism is

Surface area of a prism = 2A+PD

where,

- A = Area of the cross section
- P = Perimeter of the cross section
- D = Depth of the prism

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

**Calculate the area of the cross section.****Calculate the perimeter of the cross section.****Substitute all known values into the formula and solve.****Write the answer, including the units.**

Work out the surface area of the prism.

**Calculate the area of the cross section.**

The cross section is an L-shape and so we need to split the shape into two rectangles, calculate the area of each rectangle, then add them together to get the area of the cross section.

Area A: 2\times7=14

Area B: 7\times4=28

The total area of cross-section is 14+28=42 .

The area of the cross section A=42cm^2 .

**Calculate the perimeter of the cross section.**

The perimeter of the L-shape is equal to 2\times(9+7)=32cm .

P=32cm**Substitute all known values into the formula and solve.**

Using the formula,

surface area of a prism = 2A+PD ,

when A=42 , P=32 and D=10 , we have

\[\text{SA}=2\times{42}+(32\times{10})\\
=84+320\\
=404\]

**Write the answer, including the units.**

The surface area of the prism is 404cm^2 .

Work out the surface area of the prism.

**Calculate the area of the cross section.**

We can split the cross section into two polygons, a rectangle and a trapezium. We then work out the area of both, and add the values together to get the area of the cross section.

Area A: 5\times2=10

Area B: \frac{1}{2}(2+5) \times 4=14

10+14=24The area of the cross section A=24cm^2 .

**Calculate the perimeter of the cross section.**

The perimeter of the cross section is equal to the sum of the edges of the cross section.

P=5+2+5+2+6=20cm**Substitute all known values into the formula and solve.**

Using the formula,

Surface area of a prism = 2A+PD ,

when A=24, P=20 and D=9 , we have

\[\text{SA}=2\times{24}+(20\times{9})\\
=48+180\\
=228.\]

**Write the answer, including the units.**

The surface area of the prism is 228cm^2 .

Work out the surface area of the prism.

**Calculate the area of the cross section.**

The cross section is a trapezium with an area formula of \text{A}=\frac{1}{2}(a+b)h .

\[A=\frac{1}{2}(a+b)h\\
=\frac{1}{2}(10+4)\times{4}\\
=28\]

A=28cm^2 .

**Calculate the perimeter of the cross section.**

The perimeter of the trapezium is the sum of the edges, P=10+5+4+5=24cm .

**Substitute all known values into the formula and solve.**

Using the formula,

Surface area of a prism = 2A+PD ,

when A=28, P=24 and D=12 , we have

\[\text{SA}=2\times{28}+(24\times{12})\\
=56+288\\
=344.\]

**Write the answer, including the units.**

The surface area of the trapezoidal prism is 344cm^2 .

**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.

**Calculating with different units**

You need to make sure all measurements are in the same units before calculating surface area. (You canβt have some in cm and some in m, for example).

**Using the wrong formula**

Be careful to apply the correct prism related formula to the correct question type.

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

288cm^2

336cm^2

384cm^2

408cm^2

Calculating the area of each face, we have

Face |
Area |

Front | \frac{1}{2} Γ 8 Γ 6 = 24 |

Back | 24 |

Bottom | 12 Γ 8 = 96 |

Left side | 12 Γ 6 = 72 |

Right side | 12 Γ 10 = 120 |

Total surface area =24+24+96+72+120=336cm^{2} .

2. Work out the surface area of this prism.

154mm^2

113mm^2

226mm^2

336mm^2

Work out the area of each face,

Face |
Area |

Front | 7 Γ 2 = 1 |

Back | 14 |

Bottom | 7 Γ 11 = 77 |

Top | 77 |

Left side | 2 Γ 11 = 22 |

Right side | 22 |

Total surface area =14+14+77+77+22+22=226\mathrm{mm}^{2} .

3. Work out the surface area of this prism.

252cm^2

330cm^2

205cm^2

246cm^2

Work out the area of each face,

Face |
Area |

Front | \frac{1}{2} Γ 8 Γ 10.5 = 42 |

Back | 42 |

Top left | 5 Γ 6 = 30 |

Top right | 30 |

Bottom left | 8.5 Γ 6 = 51 |

Bottom right | 51 |

Total surface area =42+42+30+30+51+51=246\mathrm{cm}^{2} .

4. Work out the surface area of this prism in cubic millimetres.

2658mm^2

7920mm^2

133.5mm^2

833.4mm^2

First we need to convert all of the units to millimetres.

2.4cm=24mm ,

2.5cm=25mm ,

3cm=30mm .

Work out the area of each face,

Face |
Area |

Front | \frac{1}{2}(15+7) Γ 24 = 264Β |

Back | 264 |

Top | 25 Γ 30 = 750 |

Bottom | 24 Γ 30 = 720 |

Left side | 15 Γ 30 = 450 |

Right side | 7 Γ 30 = 210 |

Total surface area =264+264+750+720+450+210=2658\mathrm{mm}^{2} .

5. Work out the surface area of this prism,

742cm^2

462cm^2

1020cm^2

742cm^2

Area A: 5 \times 6=30

Area B: 7 \times 3=21

The total area of cross-section is 30+21=51 .

Work out the area of each face,

Face |
Area |

Front | 51 |

Back | 51 |

Bottom | 7 ΓΒ 20 = 140 |

Left side | 9 ΓΒ 20 = 180 |

Top 1 | 5 ΓΒ 20 = 100 |

Right side 1 | 6 ΓΒ 20 = 120 |

Top 2 | 2 ΓΒ 20 = 40 |

Right side 2 | 3 ΓΒ 20 = 60 |

Total surface area =51+51+140+180+100+120+40+60=742\mathrm{cm}^{2}

6. Work out the surface area of the prism.

3210m^2

3570m^2

11700m^2

3330m^2

Area A: 24 \times 30=720

Area B: \frac{1}{2} \times 24 \times 5=60

The total area of cross-section is 720+60=780 .

Work out the area of each face,

Face |
Area |

Front | 780 |

Back | 780 |

Left side | 30 Γ 15 = 450 |

Top | 24 Γ 15 = 360 |

Right side | 30 Γ 15 = 450 |

Right bottom | 13 Γ 15 = 195 |

Left bottom | 13 Γ 15 = 195 |

Total surface area =780+780+450+360+450+195+195=3210\mathrm{m}^{2}

1. Work out the surface area of the prism.

**(3 marks)**

Show answer

Area of cross-section =3 \times 3 + 10 \times 6=69

**(1)**

The areas of rectangular faces are 180, 162, 54, 54, 126 and 108 .

**(1)**

The total surface area is 69+69+180+162+54+54+126+108=822\mathrm{cm}^{2}

**(1)**

2. The two prisms shown have the same volume. Vicky says that means they also have the same surface area. Is Vicky correct? Show how you decide.

**(3 marks)**

Show answer

The cuboid surface area is 6+6+20+20+30+30=112\mathrm{cm}^{2} .

**(1)**

The triangular prism surface area is Β 6+6+30+40+50=132\mathrm{cm}^{2} .

**(1)**

No they do not have the same surface area.

**(1)**

3. Frank wants to paint his rabbitβs hutch. The dimensions of the hutch are shown below.

The paint that Frank wants to use covers 10 square metres per litre.The paint comes in 100ml, 500ml or 1l tins.

Frank wants to buy the smallest tin possible.

Which tin should Frank buy? Show your working.

**(4 marks)**

Show answer

The area of cross-section is \frac{1}{2}(0.5+0.75) \times 0.6=0.375 .

**(1)**

The areas of other faces are 0.9, 1.125, 0.975, 0.75 .

**(1)**

The total surface area is 0.375+0.375+0.9+1.125+0.975+0.75=4.5 .

**(1)**

Needs to cover 4.5 square metres so need 0.45l – Frank needs to buy 500ml .

**(1)**

You have now learned how to:

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

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