# Similar Figures

Two surfaces or solids are similar if any two corresponding sides or planes are proportional.

In similar figures of any kind, pairs of corresponding line segments such as x_{1}, x_{2} and y_{1}, y_{2} have the same ratio.

$\dfrac{x_1}{x_2} = \dfrac{y_1}{y_2}$

The areas of similar surfaces A_{1} and A_{2} have the same ratio as the squares of any two corresponding lines x_{1} and x_{2}.

$\dfrac{A_1}{A_2} = \dfrac{{x_1}^2}{{x_2}^2}$

The volumes of similar solids V_{1} and V_{2} have the same ratio as the cubes of two corresponding lines x_{1} and x_{2}.

$\dfrac{V_1}{V_2} = \dfrac{{x_1}^3}{{x_2}^3}$

**Some Facts About Similar Figures**

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- Regular polygons of the same kind are all similar.
- All circles are similar.
- All squares are similar.
- All equilateral triangles are similar.
- Two isosceles triangles are only similar if they have equal vertex angle.
- Right circular cones are similar if they have equal vertex angle.
- If the central angle of two circular sectors are equal, they are similar.