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Properties of shapes

Similar shapes

Similar shapes

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Explainer Video

Tutor: Labib

Summary

Similar shapes

In a nutshell

Similar shapes are shapes that are the same but different in size. This difference in size is based on a common scale factor. Shapes have to abide by at least one condition to prove they are similar. 



Similarity

Two shapes are similar if all the angles are the same or all the sides have been enlarged or reduced by the same scale factor. 


Example 1

Show that the two triangles below are similar.


Maths; Properties of shapes; KS3 Year 7; Similar shapes


Both triangles have a right-angle, and a common angle of 30∘30^\circ30∘. 

Find the missing angles using the fact that angles in a triangle add up to 180∘180^\circ180∘.  

 180−(30+90)=180−(120)=60∘180-(30+90)=180-(120)=60^\circ180−(30+90)=180−(120)=60∘.


The angles in both triangles are 90∘, 30∘90^\circ, \ 30^\circ 90∘, 30∘ and 60∘60^\circ60∘


The triangles are similar by AAA.


Note: The AAAAAAAAA rule means all three angles in a triangle are the same. 

​


Using scale factors

When two shapes are similar, it means the corresponding side lengths are proportional to one another. This means that each side in one shape can be multiplied by one number to give the corresponding side lengths in a similar shape. This number is the scale factor. 


Example 2

The two triangles shown below are similar. What is the length of the missing side?


Maths; Properties of shapes; KS3 Year 7; Similar shapes


Similar triangles will have a scale factor between them.

Find the scale factor by comparing the corresponding sides. 

6÷3=26 \div 3 = 26÷3=2​​​


The scale factor to go from the smaller to larger triangle is 222. 

The missing side is corresponding with the side with length 555. 

5×2=105 \times 2 = 105×2=10​​​

​​

Therefore, the missing side is 10‾\underline{10}10​ units long.


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FAQs - Frequently Asked Questions

How do you find the scale factor?

Compare the corresponding side lengths of both shapes and divide them to find the scale factor to go from one shape to another.

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

To prove that two triangles are similar, use the AAA rule which means all three angles are matching.

What does it mean for two shapes to be similar?

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

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