# Similarity Two shapes are **similar** if one can be obtained from the other by a series of rotations, reflections, translations, and scaling. If two shapes are similar then the following hold: * Corresponding angles on the two shapes are equal. * There is a fixed scale factor such that corresponding lengths on the two shapes are related by multiplying by this scale factor. * The ratio of two lengths in one shape is the same as the ratio of the corresponding lengths in the other. ## Similar Triangles When the two shapes are triangles then the following statements are equivalent: * The two triangles are similar * The angles in one triangle are the same as the angles in the other * The ratios of the lengths in one triangle are the same as in the other ## Scale Factors When two shapes are similar, corresponding lengths are related by a fixed **scale factor**. Corresponding areas are also related by a scale factor, and if the shape is three dimensional so are volumes. These three scale factors are related as follows: If the length scale factor is $k$, then the area scale factor is $k^2$, and the volume scale factor is $k^3$. [[!redirects similar triangle]] [[!redirects similar triangles]] [[!redirects similar shape]] [[!redirects similar shapes]] [[!redirects similar]] [[!redirects scale factor]] [[!redirects length scale factor]] [[!redirects area scale factor]] [[!redirects volume scale factor]] [[!redirects scale factors]] [[!redirects length scale factors]] [[!redirects area scale factors]] [[!redirects volume scale factors]]