Similar Triangles Calculator

Enter the three sides of a reference triangle and one corresponding side of a similar triangle to find the scale factor, the two missing sides, and the area ratio.

Quick Facts

Similarity rule
Corresponding sides are proportional
a₂/a₁ = b₂/b₁ = c₂/c₁ = k, the scale factor.
Perimeter ratio
P₂ = k × P₁
Perimeters scale by the same factor k as the sides.
Area ratio
A₂ = k² × A₁
Areas scale by the square of the linear scale factor.

Your Results

Calculated
Scale Factor (k)
-
k = a′ ÷ a
Side b′
-
b × k
Side c′
-
c × k
Area Ratio (k²)
-
Triangle 2 area ÷ Triangle 1 area

Ready

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

Formula and Method for Similar Triangles

Two triangles are similar when they have the same shape but not necessarily the same size: every pair of corresponding angles is equal, and every pair of corresponding sides is in the same ratio. That common ratio is called the scale factor, usually written k. Similarity can be confirmed with AA (two angles equal), SAS (two sides proportional with the included angle equal), or SSS (all three sides proportional). This calculator uses the SSS relationship: once you know all three sides of one triangle and a single corresponding side of a similar triangle, every other measurement of the second triangle follows automatically.

How the calculation works

Label the reference triangle's sides a, b, and c, and let a′ be the side of the second triangle that corresponds to side a. The scale factor is k = a′ ÷ a. Because corresponding sides of similar triangles are always in the same proportion (a′/a = b′/b = c′/c = k), the remaining two sides of the second triangle are found by multiplying: b′ = k × b and c′ = k × c. The same factor k scales the perimeter (P′ = k × P), while the area scales by k squared (A′ = k² × A), since area is a product of two linear dimensions.

Common mistakes

  • Matching the wrong sides: the scale factor only works if side a′ truly corresponds to side a — mismatched vertices give a meaningless ratio. Corresponding sides are the ones opposite equal angles.
  • Mixing units: enter all four side lengths in the same unit; convert first if your measurements were taken in different units.
  • Forgetting the area exponent: a common error is scaling area by k instead of k² — doubling every side (k = 2) quadruples the area (k² = 4), not doubles it.
  • Ignoring the triangle inequality: the three sides you enter for Triangle 1 must actually form a triangle — each side must be shorter than the sum of the other two.

Real-world applications

  • Indirect measurement — using a similar triangle formed by shadows or a mirror to find the height of a tree, flagpole, or building without climbing it.
  • Scale drawings, blueprints, and maps, where every distance is a real-world length multiplied by a fixed scale factor.
  • Photography and projection, where an image is enlarged or reduced while keeping every proportion — and therefore every angle — the same.
  • Engineering and model-building, where a small-scale model must have all linear dimensions related to the full-size object by one consistent k.

Frequently Asked Questions

What does it mean for two triangles to be similar?
Similar triangles have the same shape but not necessarily the same size: their corresponding angles are equal and their corresponding sides are all in the same proportion. Similarity can be established by AA (two equal angles), SAS (two proportional sides with the included angle equal), or SSS (all three sides proportional).
How do I find the scale factor between two similar triangles?
Divide one side of the second triangle by the corresponding side of the first triangle: k = a′ ÷ a. Once you know k, multiply every other side of the first triangle by k to get the matching sides of the second triangle.
How does the area of similar triangles relate to the scale factor?
The area ratio equals the square of the linear scale factor: A′ = k² × A. For example, if the sides double (k = 2), the area becomes 4 times as large (k² = 4), while the perimeter only doubles.
Can I find the missing sides of a similar triangle from just one known side?
Yes. If you know all three sides of the first triangle and one side of the second triangle that corresponds to a known side of the first, you can compute the scale factor k and then find the remaining two sides by multiplying each corresponding side of the first triangle by k.

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