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Surface area of 3D shapes: Nets, formulae

Surface area of 3D shapes: Nets, formulae

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Summary

Surface area of 3D shapes: Nets, formulae

​​In a nutshell

The surface area of a shape is the sum of the areas of its faces. The surface area of 3D shapes can either be calculated using nets or using given formulae.



Surface area using nets

Net

A net is a 3D shape that has been unfolded and laid out flat. Here are some examples of nets.



SHAPE

NET

Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae

Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae

Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae

Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae

Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae

Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae



Surface area

To find the surface area of a shape using nets, sketch the net and work out the area of the net. This can be done by working out the area of each individual shape in the net.


Example 1

What is the surface area of a cube with side lengths 8cm8cm8cm?


First, sketch the net:


Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae


The net of a cube is six squares. In this example, each square will have side lengths 8cm8cm8cm.


Then, find the area of the net:

The net is a compound shape consisting of six squares.

The area of one of these squares is given to be:

8×8=64cm28\times8=64cm^28×8=64cm2​​


The area of the net is therefore:

​6×64=384cm26\times64=384cm^26×64=384cm2​​


The surface area of the cube is 384 cm2‾\underline{384 \ cm^2}384 cm2​.


​

Surface area formulae

These are the formulae you need to know:


SHAPE

FORMULA

DIAGRAM

Sphere

​S.A=4πr2S.A=4\pi r^2S.A=4πr2​​
Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae

Cylinder

​S.A=2πr2+2πrhS.A=2\pi r^2 +2\pi rhS.A=2πr2+2πrh​​
Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae

Cone
​Curved surface area=πrl\text{Curved surface area}=\pi rlCurved surface area=πrl​​
​S.A=πrl+πr2S.A=\pi rl +\pi r^2S.A=πrl+πr2​​
Maths; Shapes and area; KS4 Year 10; Surface area of 3D shapes: Nets, formulae


Example 2

What is the surface area of a sphere with radius 2m2m2m to three significant figures?


Substitute r=2r=2r=2 into the formula:

​S.A=4πr2=4π(2)2=16π=50.26548246...m2\begin{aligned}S.A&=4\pi r^2\\&=4\pi (2)^2\\&=16\pi\\&=50.26548246...m^2\end{aligned}S.A​=4πr2=4π(2)2=16π=50.26548246...m2​​​


The surface area is 50.3m2  (3 s.f.)‾\underline{50.3m^2\space\ (3\space s.f.)}50.3m2  (3 s.f.)​.


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Surface area of 3D shapes: Nets, formulae

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FAQs - Frequently Asked Questions

Which shapes have a formula to work out their surface area?

There are given formulae to work out the surface area of a sphere, a cylinder and a cone.

How do you work out the surface area of a shape using nets?

First, sketch the net. Then, work out the area of the net by working out the area of each individual shape in the net.

What are the two ways of working out surface area?

The two ways of finding the surface area of a shape is to use a net or to use a formula.

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