Similar Figures

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

Example of similar figures: spheres, cylinders, and regular hexagons

 

In similar figures of any kind, pairs of corresponding line segments such as x1, x2 and y1, y2 have the same ratio.

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

 

The areas of similar surfaces A1 and A2 have the same ratio as the squares of any two corresponding lines x1 and x2.

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

 

The volumes of similar solids V1 and V2 have the same ratio as the cubes of two corresponding lines x1 and x2.

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

 

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.

 

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