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Similar shapes: Scaling

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Similar shapes: Scaling

In a nutshell

Similar shapes are shapes that are the same shape, but have different sizes. There is a single condition to prove two shapes are similar. Similar shapes have a scale factor which can be used to find missing lengths.


Two shapes are similar if they have the same angles. This means that the condition to prove that two triangles are similar is AAA - three common angles. Two similar shapes can have different side lengths that are in proportion to one another.

Example 1

Show that the two triangles below are similar.

Maths; Shapes and area; KS4 Year 10; Similar shapes: Scaling

Both triangles have a right-angle, and a common angle of 3030^\circ. This means they also have a common angle of 180(30+90)=60180-(30+90)=60^\circ.

The triangles are similar due to AAA.

Similarity and scaling

When two shapes are similar, it means the matching side lengths are proportional to one another. This means that the side lengths of one shape can be multiplied by a scale factor to give the side lengths of another similar shape.

Example 2

The two triangles shown below are similar. What is the value of xx?

Maths; Shapes and area; KS4 Year 10; Similar shapes: Scaling

If the two triangles are similar, then there must be a common scale factor.

Find the scale factor by comparing the lengths RMRM and LSLS:


The scale factor to go from triangle AMRAMR to triangle LOSLOS is therefore 22.

Use this scale factor to find the value of xx by focusing on the lengths ARAR and LOLO:

LO=2×ARLO=2\times AR​​

3x+3=2×(x+2)3x+3=2\times (x+2)​​



Therefore, x=1\underline{ x=1}.

Want to find out more? Check out these other lessons!

Similar shapes

Frequently Asked Questions (FAQ)


  • Question: How do you use similarity to find the length of a missing side?

    Answer: Compare corresponding lengths to find a scale factor, then use the scale factor to find the missing length.

  • Question: What is the condition to prove that two triangles are similar?

    Answer: To prove two triangles are similar, use AAA - three matching angles.

  • Question: What does it mean for two shapes to be similar?

    Answer: Two shapes are similar if they have the same shape but have scaled side lengths.



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