What Are Similar Shapes?

Similar Shapes

Two shapes are similar if one is an enlargement (or reduction) of the other. They have the same shape but different sizes.

When two figures are similar:

The figures are equiangular — all corresponding angles are equal
The corresponding sides are in proportion — they share the same ratio

This ratio is called the scale factor (often written as $k$ ).

Analogy

Think of similar shapes like a photo on your phone that you zoom in or out on. The image looks the same — same proportions, same angles — it's just bigger or smaller. Nothing gets stretched or distorted.

For example, if triangle $A B C$ is similar to triangle $P QR$ , then:
$\frac{P Q}{A B} = \frac{QR}{B C} = \frac{R P}{C A} = k$

where $k$ is the enlargement factor (scale factor).

What Are Similar Shapes?

The Scale Factor (k)

Scale Factor (k)

The scale factor $k$ is the ratio of any length in the image (enlarged or reduced shape) to the corresponding length in the original shape. $k = \frac{length in image}{corresponding length in original}$

If $k > 1$ , the shape has been enlarged (made bigger)
If $0 < k < 1$ , the shape has been reduced (made smaller)
If $k = 1$ , the shapes are congruent (identical in size)

Example

A 10 cm by 8 cm photograph is enlarged so that the longer side becomes 25 cm.

Finding $k$ :
$k = \frac{new length}{original length} = \frac{25}{10} = 2.5$

Finding the shorter side:
$new shorter side = k \times 8 = 2.5 \times 8 = 20 cm$

The enlarged photo is 25 cm by 20 cm.

Warning

Make sure you match corresponding sides when calculating $k$ . The longest side of one shape must be compared with the longest side of the other, and so on.

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