top of page

19.1 Similarity and Enlargement

GCSE Maths Foundation · Edexcel · Chapter 19: Congruence, Similarity and Vectors

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: similar shapes, scale factors and finding missing lengths. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Decide whether two shapes are similar
  • Work out the scale factor between two similar shapes
  • Use the scale factor to find missing lengths in similar shapes
  • Explain why an enlargement always gives a similar shape

Key words

  • Similar: Two shapes are similar if one is an enlargement of the other. They have the same shape but can be different sizes.
  • Corresponding: Sides or angles that are in the same position in the two shapes. They match up.
  • Scale factor: The number you multiply every length by to get from one shape to the other.
  • Enlargement: A transformation that changes the size of a shape. Every length is multiplied by the scale factor.

What makes shapes similar?

Two shapes are similar when:

  • all the corresponding angles are equal, and
  • all the corresponding sides are in the same ratio (each side is multiplied by the same scale factor).

Worked example: are these rectangles similar?

Rectangle P is 4 cm by 6 cm. Rectangle Q is 10 cm by 15 cm. Rectangle R is 6 cm by 8 cm.
P to Q: 10 ÷ 4 = 2.5 and 15 ÷ 6 = 2.5. Same scale factor, so P and Q are similar.
P to R: 6 ÷ 4 = 1.5 but 8 ÷ 6 = 1.33… Different, so P and R are not similar.

Answer: P and Q are similar; P and R are not

Key fact: All rectangles have four 90° angles, but they are not all similar. The sides must also be in the same ratio.

Scale factors and missing lengths

Find a pair of corresponding sides where you know both lengths. Then:

  • 1; What to do Scale factor = length on new shape ÷ matching length on old shape
  • 2; What to do Small → big: multiply by the scale factor
  • 3; What to do Big → small: divide by the scale factor

Worked example: finding two missing sides

Scale factor = 10 ÷ 4 = 2.5
x matches BC: x = 7 × 2.5 = 17.5 cm
y matches AC: y = 5 × 2.5 = 12.5 cm

Answer: x = 17.5 cm, y = 12.5 cm

Worked example: going from big to small

Two similar kites. A side of 6 cm on the small kite matches a side of 21 cm on the large kite.
Another side of the large kite is 14 cm. Find the matching side on the small kite.
Scale factor = 21 ÷ 6 = 3.5. Small side = 14 ÷ 3.5 = 4 cm

Answer: 4 cm

Similar triangles from angles

For triangles there is a shortcut: if all three angles of one triangle match the three angles of another, the triangles are similar. You only need two angles, because the third is 180° minus the other two.

Worked example: checking the angles

Triangle 1 has angles 48° and 67°. Triangle 2 has angles 67° and 65°.
Third angle of triangle 1 = 180° − 48° − 67° = 65°
So both triangles have angles 48°, 65° and 67°.

Answer: The triangles are similar

Key fact: Angles never change in an enlargement. Only the lengths are multiplied by the scale factor.

Enlargement makes similar shapes

When you enlarge a shape, every length is multiplied by the same scale factor and the angles stay the same. So the image is always similar to the original.

Worked example: perimeter of an enlargement

A triangle has sides 3 cm, 4 cm and 6 cm. It is enlarged by scale factor 5.
New sides: 15 cm, 20 cm, 30 cm. Perimeter = 15 + 20 + 30 = 65 cm
Quicker: old perimeter 13 cm × 5 = 65 cm

Answer: 65 cm

Common mistakes

  • Adding instead of multiplying: if one side goes from 4 cm to 10 cm, the others do not all get 6 cm longer.
  • Matching the wrong sides. Use the angles or the order of the letters to find corresponding sides.
  • Multiplying when you should divide: going from the big shape to the small shape, divide by the scale factor.
  • Thinking the angles get bigger too. In similar shapes the angles are always the same.

Revision questions

Try these, then check your answers below each one.

Question 1. A rectangle 4 cm by 7 cm is enlarged to make a similar rectangle. The short side of the new rectangle is 12 cm. How long is its long side?

Answer: 21 cm

Working: Scale factor = 12 ÷ 4 = 3. Long side = 7 × 3 = 21 cm.

Question 2. Which rectangle is similar to a rectangle 6 cm by 9 cm? (Choose from: 8 cm by 12 cm, 8 cm by 11 cm, 9 cm by 12 cm, 10 cm by 13 cm.)

Answer: 8 cm by 12 cm

Working: 9 ÷ 6 = 1.5 and 12 ÷ 8 = 1.5. The others give 11 ÷ 8, 12 ÷ 9 and 13 ÷ 10, which are not 1.5.

Question 3. Two triangles are similar. A side of 6 cm on the small triangle matches a side of 15 cm on the large triangle. What is the scale factor from small to large?

Answer: 2.5

Working: 15 ÷ 6 = 2.5

Related lessons

Continue in the full course

This lesson is section 19.1 (EGCMF1901) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

bottom of page