Geometry I: Similar Shapes – JSS3 Mathematics Lesson Note (Second Term)

Similar shapes are shapes that have the same form and equal angles but different sizes, with their sides in the same ratio.

Behavioural objectives

By the end of this lesson, students should be able to:

1. Define similar shapes in simple words.
2. State the two conditions for shapes to be similar.
3. Find the scale factor between two similar shapes.
4. Tell the difference between similar shapes and congruent (same size) shapes.
5. Find similar shapes around them and use them to solve simple problems.

What are similar shapes?

Similar shapes are shapes that look the same but are different sizes.

• Same form: a triangle is similar only to another triangle, a circle only to another circle.
• Equal angles: matching corners are the same size in both shapes.
• Sides in the same ratio: every side of one shape is multiplied by the same number to give the other.
• Enlargement or reduction: one shape is the other stretched or shrunk by the same amount in every direction.

• Example: a photograph and its enlarged print look alike, but the print is bigger.

What is the scale factor?

The scale factor is the number you multiply each side of one shape by to get the matching side of the similar shape.

• Enlargement: the scale factor is greater than 1, for example 2 or 3.
• Reduction: the scale factor is less than 1, for example 1/2.
• Formula: scale factor = length of a side in the new shape ÷ length of the matching side in the old shape.

What are examples of similar shapes?

Triangles

• Angles: a big triangle and a small triangle are similar when their three angles match.
• Sides: the sides are in the same ratio, so the big triangle is the small one enlarged.

Rectangles

• Angles: all corners are 90 degrees in every rectangle, so the angles always match.
• Sides: two rectangles are similar only if length and width grow or shrink by the same factor.
• Squares: all squares are similar to each other.

Circles

• All circles: any two circles are similar, whatever their sizes.
• Reason: they have the same round shape and differ only in radius.

How do you tell if shapes are similar?

Check the form, the angles and the side ratios, in that order.

1. Look at the form. Both shapes must be the same kind, such as triangle with triangle.
2. Check the angles. Matching corners must be equal. In triangles, the three angles must match.
3. Compare the sides. Divide each side of the new shape by the matching side of the old shape.
4. Decide. If every division gives the same number, the shapes are similar.

Worked example 1: Are the rectangles similar?

• Rectangle A is 4 cm by 2 cm. Rectangle B is 6 cm by 4 cm.
• Length: 6 ÷ 4 = 1.5.
• Width: 4 ÷ 2 = 2.
• The numbers are different, so the rectangles are not similar.

Worked example 2: Find the missing side

• Triangle P has sides 3 cm, 4 cm and 5 cm. Triangle Q is similar, with sides 6 cm, 8 cm and x.
• Scale factor: 6 ÷ 3 = 2.
• x = 5 × 2 = 10 cm.

Worked example 3: Rectangle with an unknown width

• A rectangle 3 cm by 2 cm is enlarged to a similar rectangle 9 cm long.
• Scale factor: 9 ÷ 3 = 3.
• Width = 2 × 3 = 6 cm.

What is the difference between similar and congruent shapes?

Similar shapes have the same form and angles but may differ in size. Congruent shapes have the same form, angles and size.

What makes shapes not similar?

Shapes are not similar when their form, angles or side ratios do not match.

• Different forms: a triangle and a square.
• Different angles: two triangles whose angles are not equal.
• Different side ratios: the sides do not grow by the same factor.
• One-direction stretch: a shape pulled longer in only one direction.

Examples of shapes that are not similar

• Rectangles: a tall thin rectangle and a short wide rectangle.
• Triangles: two triangles with different angle sizes.
• Circle and oval: an oval (egg shape) is a stretched circle.

Where can you find similar shapes in real life?

Many objects around you are close to similar, but check the side ratios for rectangles before you say so.

At home

• Plates: plates of different sizes are all round, so they are similar.
• Screens: a TV and a phone screen are similar only if their length-to-width ratios match.
• Windows: a large and a small window are similar if their sides are in the same ratio.

At school

• Balls: balls of different sizes look like circles from the side, so they are similar.
• Boards and paper: a blackboard and a sheet of paper are rectangles, similar only if the ratios match.
• Door and notebook: both are rectangles, so measure them to check.

In nature

• Leaves: leaves of one tree in different sizes look alike in form.
• Flowers: flowers of the same type but different sizes look alike in form.
• Shells: shells of the same kind on the beach have the same form but different sizes.

How are similar shapes used in everyday life?

Similar shapes are used wherever an object must be drawn, copied or shown at a different size.

• Photography: zooming in or out gives a similar rectangle, with the same shape but a different size.
• Building and design: architects make small models that are similar to the real building.
• Art and drawing: artists enlarge a small sketch into a large painting.
• Maps: a map shows places in a shape similar to the real land, much reduced.

What activities can students do to practise?

Activity 1: Shape hunt

• Find two similar circles in the classroom.
• Find two rectangles and measure them to check if they are similar.
• Find two similar triangles, if any are available.

Activity 2: Draw similar shapes

• Draw a small triangle.
• Draw a bigger triangle with the same shape.
• Use a protractor to check that the angles are equal.

Activity 3: Size check

• Start with a rectangle 2 units long and 1 unit wide.
• Draw a similar rectangle 4 units by 2 units (scale factor 2).
• Draw a similar rectangle 6 units by 3 units (scale factor 3).

What are the important words to remember?

• Similar: having the same form and equal angles but different sizes.
• Congruent: having the same form and the same size.
• Ratio: a comparison of two numbers, such as 2 to 1.
• Scale factor: the number each side is multiplied by in an enlargement or reduction.
• Angle: the amount of turn between two lines that meet at a point, shown at a corner.
• Corresponding: matching parts, such as the longest side of one triangle and the longest side of its similar triangle.

Evaluation

1. What two conditions make two shapes similar?
2. One rectangle is 4 cm by 2 cm. Another is 8 cm by 4 cm. Are they similar? (Answer: Yes, because 8 ÷ 4 = 2 and 4 ÷ 2 = 2.)
3. Can a square and a rectangle ever be similar? (Answer: Yes, if the rectangle is also a square.)
4. Triangle A has sides 2 cm, 3 cm and 4 cm. A similar triangle B has a shortest side of 6 cm. Find its other two sides. (Answer: 9 cm and 12 cm.)
5. State one difference between similar shapes and congruent shapes. (Answer: Congruent shapes have the same size, but similar shapes may differ in size.)

Assignment

Find three pairs of shapes at home or school and test whether each pair is similar.

1. Name the shapes and measure their sides.
2. Divide matching sides to find the ratios.
3. State which pairs are similar and give the scale factor for each.