Properties of Similar Polygons

Similar Polygons

Similar polygons are shapes that have the same shape but not necessarily the same size. They maintain proportional side lengths and identical angles.

The following diagrams show the properties of similar polygons. Scroll down the page for more examples and solutions.

 

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Two polygons are similar if:

1. Corresponding angles are equal.
2. Corresponding sides are proportional.

Notation:
If polygon ABCD is similar to polygon WXYZ, we write:
ABCD ∼ WXYZ

Scale Factor
The scale factor is the ratio of any two corresponding sides in similar polygons.
If Polygon 1 has side lengths s₁, s₂, s₃, … and Polygon 2 has corresponding side lengths S₁, S₂, S₃, … then:

\(\frac{S_1}{s_1} = \frac{S_2}{s_2} = \frac{S_3}{s_3} = … = k\)

where k is the scale factor.
If k>1, Polygon 2 is an enlargement of Polygon 1.
If 0<k<1, Polygon 2 is a reduction of Polygon 1.
If k=1, the polygons are congruent.

Ratio of Perimeters
The ratio of the perimeters of two similar polygons is equal to their linear scale factor.

Ratio of Areas
The ratio of the areas of two similar polygons is equal to the square of their linear scale factor.

Similar Polygons
Two polygons are similar if their corresponding angles are congruent and the corresponding sides have a constant ratio (in other words, if they are proportional).
Typically, problems with similar polygons ask for missing sides. To solve for a missing length, find two corresponding sides whose lengths are known. After we do this, we set the ratio equal to the ratio of the missing length and its corresponding side and solve for the variable.

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Similar Figures - Similar Polygons
Students learn that similar polygons have the same shape, and if two polygons are similar, then the corresponding angles are congruent, and the corresponding sides are in proportion. Students also learn that the scale factor is ratio of the lengths of two corresponding sides.

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