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.