∑Richard Hummel Math

Similarity

When two figures have the same shape but different sizes — equal angles, proportional sides, and scale factors.

Practice exercises →
Similar triangles — same angles, proportional sides (scale factor 1.7)
originalscaled copy

Side ratios

170/100 = 1.7
184/108 = 1.7
199/117 = 1.7

Matching colored arcs mark equal angles. Side lengths change together by the same scale factor.

Definition

Two figures are similar if they have the same shape but possibly different sizes. One is an enlarged or reduced version of the other — like a photograph and a photocopy at a different zoom level.

The symbol for similarity is ∼\sim.

For two similar figures:

  • All corresponding angles are equal
  • All corresponding sides are in the same ratio (the scale factor)

If △ABC∼△DEF\triangle ABC \sim \triangle DEF with scale factor kk, then: DEAB=EFBC=FDCA=k\frac{DE}{AB} = \frac{EF}{BC} = \frac{FD}{CA} = k

Definition

The triangle similarity criteria — you need fewer conditions than for congruence:

  • AA (Angle-Angle): two pairs of equal angles (the third automatically matches)
  • SAS similarity: two sides in proportion and the included angle equal
  • SSS similarity: all three sides in proportion
Identifying similar triangles

△ABC\triangle ABC has angles 50°50°, 70°70°, 60°60°. △DEF\triangle DEF has angles 50°50°, 70°70°, 60°60°.

Two angles match (in fact all three do). By AA, △ABC∼△DEF\triangle ABC \sim \triangle DEF.

Try it

A tree casts a shadow 1515 m long. At the same time, a 22 m tall person casts a shadow 33 m long. How tall is the tree?

Solution

The person, their shadow, and the sun's ray form a triangle similar to the triangle formed by the tree, its shadow, and the same sun's ray. The scale factor is 153=5\frac{15}{3} = 5.

Tree height =5×2=10= 5 \times 2 = 10 m.

Related concepts

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Triangles
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