What Is Similarity?

How similar figures are related

For triangles, similarity can often be established by AA, SSS~, or SAS~. Once similarity is known, the figures support proportion reasoning, scale factors, and many geometric shortcuts.

This page keeps the two figures visible together so similarity is read from stable angle structure and consistent side ratios, not from a vague sense that the figures look related.

If every side in one figure is twice the matching side in another, the scale factor is 2. The angles remain the same even though the perimeter doubles and the area becomes four times as large. That difference between lengths and area is an important consequence of similarity.

A photograph enlarged on a screen is a familiar similarity example: the image may be bigger, but the angle pattern remains. If one dimension is stretched more than another, the picture no longer has one scale factor and the figures are not similar.

• Similar figures have the same shape but not necessarily the same size.
• Similar figures preserve angle measure but not necessarily side length.
• Corresponding side lengths of similar figures are proportional.
• Similarity is the natural geometric language of scaling, maps, models, and right-triangle proportionality.

Where similarity helps

• Use similarity when setting up side proportions between related triangles or polygons.
• Use it in scale-drawing, map, and model problems where size changes but shape does not.
• Use similarity in proofs, indirect measurement, and trigonometric preparation.

Similarity checks

• Do not treat similar figures as congruent if the scale factor is not 1.
• Do not match the wrong corresponding sides or angles when writing proportions.
• Do not use a proportion unless the figure match has been justified first.