SSS Similarity — Definition, Rules & Examples

SSS Similarity

Side-side-side similarity. When two triangles have corresponding sides with identical ratios as shown below, the triangles are similar.

Two triangles ABC and DEF with SSS Similarity: AB/DE = BC/EF = CA/FD, showing corresponding sides have equal ratios.

See also

Similarity tests for triangles

Key Formula

\frac{a}{d} = \frac{b}{e} = \frac{c}{f} = k \implies \triangle ABC \sim \triangle DEF

Where:

$a, b, c$ = The three side lengths of the first triangle (△ABC)
$d, e, f$ = The corresponding three side lengths of the second triangle (△DEF)
$k$ = The common ratio (scale factor) between corresponding sides

Worked Example

Problem: Triangle ABC has sides a = 6, b = 9, and c = 12. Triangle DEF has sides d = 4, e = 6, and f = 8. Determine whether the two triangles are similar by SSS Similarity.

Step 1: Match corresponding sides. Compare the shortest side to the shortest side, the middle to the middle, and the longest to the longest.

a = 6 \leftrightarrow d = 4, \quad b = 9 \leftrightarrow e = 6, \quad c = 12 \leftrightarrow f = 8

Step 2: Compute the ratio of each pair of corresponding sides.

\frac{a}{d} = \frac{6}{4} = \frac{3}{2}, \quad \frac{b}{e} = \frac{9}{6} = \frac{3}{2}, \quad \frac{c}{f} = \frac{12}{8} = \frac{3}{2}

Step 3: Check whether all three ratios are equal.

\frac{3}{2} = \frac{3}{2} = \frac{3}{2} \quad \checkmark

Step 4: Since all three ratios are equal, the SSS Similarity condition is satisfied. The triangles are similar with a scale factor of 3/2.

\triangle ABC \sim \triangle DEF \text{ with scale factor } k = \frac{3}{2}

Answer: Yes, △ABC ~ △DEF by SSS Similarity, with a scale factor of 3/2.

Another Example

This example shows what happens when the test fails — when one ratio differs from the others, the triangles are not similar. Students need to see both passing and failing cases.

Problem: Triangle PQR has sides 5, 10, and 14. Triangle XYZ has sides 7.5, 15, and 20. Are the two triangles similar by SSS Similarity?

Step 1: Order each triangle's sides from shortest to longest and pair them up.

PQR: 5, 10, 14 \quad \leftrightarrow \quad XYZ: 7.5, 15, 20

Step 2: Compute the ratio for each pair of corresponding sides.

\frac{5}{7.5} = \frac{2}{3}, \quad \frac{10}{15} = \frac{2}{3}, \quad \frac{14}{20} = \frac{7}{10}

Step 3: Compare the three ratios. The first two equal 2/3, but the third equals 7/10.

\frac{2}{3} = \frac{2}{3} \neq \frac{7}{10}

Step 4: Because the ratios are not all equal, the SSS Similarity condition is not met.

Answer: No, △PQR and △XYZ are not similar by SSS Similarity because the ratios of corresponding sides are not all equal.

Frequently Asked Questions

What is the difference between SSS Similarity and SSS Congruence?

SSS Congruence requires all three pairs of corresponding sides to be equal in length, meaning the triangles are the same size and shape. SSS Similarity only requires the three pairs of corresponding sides to be in the same ratio, so the triangles have the same shape but can differ in size. Congruence is a special case of similarity where the scale factor equals 1.

How do you know which sides are corresponding in SSS Similarity?

If the vertex correspondence is not given, sort each triangle's sides from shortest to longest. Then pair the shortest with the shortest, the middle with the middle, and the longest with the longest. Corresponding sides are opposite corresponding angles, so the pairing must be consistent across all three pairs.

Can you use SSS Similarity if you only know two pairs of sides?

No. SSS Similarity requires all three pairs of corresponding side ratios to be equal. If you know only two sides and need to prove similarity, you may be able to use SAS Similarity (two sides in proportion with the included angle equal) instead.

SSS Similarity vs. AA Similarity

	SSS Similarity	AA Similarity
What you need	All three pairs of corresponding sides	Two pairs of corresponding angles
Condition	Ratios of all three corresponding side pairs are equal	Two corresponding angles are equal (third is automatic)
When to use	When you know side lengths but not angle measures	When you know angle measures but not all side lengths
Scale factor	Directly found from the side ratios	Not determined unless side lengths are also given

Why It Matters

SSS Similarity appears frequently in geometry courses when you need to prove two triangles are similar using side lengths alone, without angle information. It is essential for solving problems involving indirect measurement, such as finding the height of a building using shadows or determining unknown distances on a map. Many standardized tests and proof-based exercises require you to identify which similarity criterion (SSS, SAS, or AA) applies to a given pair of triangles.

Common Mistakes

Mistake: Comparing sides in the wrong order — for example, dividing the shortest side of one triangle by the longest side of the other.

Correction: Always pair corresponding sides correctly: shortest with shortest, middle with middle, longest with longest. If vertex labels are given, match sides opposite the same-named vertices.

Mistake: Concluding similarity after checking only two of the three side ratios.

Correction: SSS Similarity requires all three ratios to be equal. Two matching ratios are not sufficient — you must verify the third ratio as well. If you only have two sides and an included angle, use SAS Similarity instead.

Related Terms

Similar — General definition of similar figures
Similarity Tests for Triangles — Overview of all triangle similarity criteria
Triangle — The geometric shape this test applies to
Corresponding — Matching sides and angles between figures
Ratio — The comparison used to check proportionality
Side of a Polygon — The line segments being compared
Scale Factor — The common ratio between similar figures