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Similar — Definition, Properties & Examples

Similar

Identical in shape, although not necessarily the same size.

 

Two similar irregular blob shapes: a smaller one on the left and a larger one on the right, identical in form but different in...

 

 

See also

Scale factor, congruent, similarity tests for triangles

Key Formula

a1a2=b1b2=c1c2=k\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = k
Where:
  • a1,b1,c1a_1, b_1, c_1 = Side lengths of the first figure
  • a2,b2,c2a_2, b_2, c_2 = Corresponding side lengths of the second figure
  • kk = Scale factor (the constant ratio between corresponding sides)

Worked Example

Problem: Triangle ABC has sides 3 cm, 4 cm, and 5 cm. Triangle DEF has sides 6 cm, 8 cm, and 10 cm. Are the two triangles similar? If so, find the scale factor.
Step 1: Match the sides in order from shortest to longest and compute the ratio of each pair of corresponding sides.
63=2,84=2,105=2\frac{6}{3} = 2, \quad \frac{8}{4} = 2, \quad \frac{10}{5} = 2
Step 2: Check whether all three ratios are equal.
2=2=2✓2 = 2 = 2 \quad \checkmark
Step 3: Since all corresponding side ratios are equal, the triangles are similar. The symbol for similarity is ~.
△ABC∼△DEF\triangle ABC \sim \triangle DEF
Answer: Yes, the triangles are similar with a scale factor of 2 (triangle DEF is twice the size of triangle ABC).

Another Example

Problem: Rectangle P measures 4 cm by 6 cm. Rectangle Q measures 6 cm by 9 cm. Are they similar?
Step 1: All rectangles have four 90° angles, so the angle condition is automatically satisfied. Check the ratio of corresponding sides (width to width, length to length).
64=1.5,96=1.5\frac{6}{4} = 1.5, \quad \frac{9}{6} = 1.5
Step 2: Both ratios are equal, so the rectangles are similar with scale factor 1.5.
k=1.5k = 1.5
Answer: Yes, Rectangle Q is similar to Rectangle P with a scale factor of 1.5.

Frequently Asked Questions

What is the difference between similar and congruent?
Similar figures have the same shape but can be different sizes — their corresponding sides are proportional. Congruent figures have both the same shape and the same size, meaning every corresponding side and angle is exactly equal. Congruence is actually a special case of similarity where the scale factor is 1.
Are all circles similar to each other?
Yes. Every circle has the same shape; they differ only in radius. You can always scale one circle to match another, so any two circles are similar. The same is true for any two squares and any two equilateral triangles.

Similar vs. Congruent

Similar figures have the same shape with corresponding sides in a constant ratio (scale factor kk). Congruent figures have the same shape AND the same size, which means k=1k = 1. Every congruent pair is also similar, but not every similar pair is congruent.

Why It Matters

Similarity is foundational in geometry and real-world applications like maps, scale models, and engineering blueprints, where objects must be proportionally resized. It also underpins trigonometry: the reason sine, cosine, and tangent depend only on angle measure is that all right triangles sharing an acute angle are similar. Understanding similarity lets you find unknown lengths in figures that would be impractical to measure directly.

Common Mistakes

Mistake: Assuming that equal angles alone prove similarity for non-triangles.
Correction: For triangles, matching angles is sufficient (AA criterion). But for other polygons, you need both equal corresponding angles AND proportional corresponding sides. For example, a 1×3 rectangle and a 1×1 square both have all 90° angles, but they are not similar.
Mistake: Mixing up the order of corresponding vertices when writing the similarity statement.
Correction: The order matters. Writing △ABC ~ △DEF means A corresponds to D, B to E, and C to F. Scrambling the order leads to incorrect side pairings and wrong calculations.

Related Terms