Solve Similar Triangles Calculator

Enter Triangle 1's three sides and one known side of a similar Triangle 2, and this tool solves for Triangle 2's remaining sides, scale factor, perimeter, and area ratio using a′/a = b′/b = c′/c.

Quick Facts

Similarity rule
a′/a = b′/b = c′/c = k
Corresponding sides of similar triangles share one scale factor, k.
Perimeter ratio
P2 = P1 × k
Perimeter scales linearly with the scale factor.
Area ratio
Area2 = Area1 × k²
Area scales with the square of the scale factor.

Your Results

Calculated
Scale factor (k)
-
k = a′ / a
Triangle 2 — Side b′
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b′ = b × k
Triangle 2 — Side c′
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c′ = c × k
Triangle 2 perimeter & area ratio
-
P2 = P1 × k, Area ratio = k²

Ready

Enter Triangle 1's three sides and one known side of Triangle 2, then press Calculate.

How to Solve Similar Triangles

Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. That single proportion is called the scale factor, k. If Triangle 1 has sides a, b, c and Triangle 2 has the corresponding sides a′, b′, c′, then a′/a = b′/b = c′/c = k. Once you know any one pair of corresponding sides, you can find the scale factor and use it to solve for every other side of the second triangle.

The proportionality method, step by step

Given a fully known Triangle 1 (sides a, b, c) and a single known side of Triangle 2 that corresponds to side a, the calculator finds the scale factor first: k = a′ / a. It then applies that same factor to the other two sides: b′ = b × k and c′ = c × k. Because perimeter is a linear sum of the sides, the perimeter also scales by k (Perimeter2 = Perimeter1 × k), while area scales by k² (Area2 = Area1 × k²) since area depends on two linear dimensions multiplied together.

Checking your triangle is valid

  • Triangle inequality: any three side lengths only form a real triangle if the sum of the two shorter sides exceeds the longest side (a + b > c, a + c > b, b + c > a). The calculator checks this for Triangle 1 before solving.
  • Similarity criteria (AA, SAS, SSS): in practice you confirm two triangles are similar via Angle-Angle (two equal angle pairs), Side-Angle-Side (two proportional sides with the included angle equal), or Side-Side-Side (all three sides proportional). This calculator assumes similarity is already established and focuses purely on solving the proportion.
  • Scale factor direction: k > 1 means Triangle 2 is larger than Triangle 1; 0 < k < 1 means it is smaller. A scale factor of exactly 1 means the triangles are congruent, not just similar.

Common uses

Similar triangles show up in map and blueprint scaling, indirect height measurement (using shadows or a mirror), photography and projection scaling, engineering scale models, and countless geometry proofs. Whenever one triangle is a scaled copy of another, this same a′/a = b′/b = c′/c relationship lets you solve for any missing length.

Frequently Asked Questions

What does it mean for two triangles to be similar?
Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same proportion. This proportion is the scale factor k, so if Triangle 1 has sides a, b, c and Triangle 2 has corresponding sides a′, b′, c′, then a′/a = b′/b = c′/c = k.
How do I find the scale factor between two similar triangles?
Divide one side of the second triangle by its corresponding side on the first triangle: k = a′ / a. Once you know k, multiply every other side of Triangle 1 by k to get the matching side of Triangle 2 (b′ = b × k, c′ = c × k).
How does the scale factor affect perimeter and area?
Perimeter scales linearly with the scale factor: Perimeter2 = Perimeter1 × k. Area scales with the square of the scale factor: Area2 = Area1 × k². So doubling the side lengths (k = 2) doubles the perimeter but quadruples the area.
What if the three given sides don't form a valid triangle?
Every triangle must satisfy the triangle inequality: the sum of any two sides must be greater than the third side (a + b > c, a + c > b, and b + c > a). If your Triangle 1 sides fail this test, no such triangle — similar or otherwise — can exist, and the calculator will flag it as invalid.

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