Similar Triangles Calculator

Background.

A fence post 1.8 m tall throws a shadow 2.4 m long. At the same moment, a tree nearby throws one 16 m long. Sunlight reaches both at the same angle, so the post, its shadow and the line of the sun's ray form a triangle with exactly the same angles as the tree, its shadow and its ray. The larger triangle is the smaller one scaled up by 16 ÷ 2.4, about 6.6667, and the tree is 1.8 × 6.6667 = 12 m tall. Nobody had to climb it.

That is the whole idea of similar triangles: the same shape at a different size. Euclid proved in Book VI of the Elements that two descriptions of it are equivalent. Triangles with equal angles have proportional sides (Proposition 4), and triangles with proportional sides have equal angles (Proposition 5). The second result means six lengths are enough to settle whether two triangles match, without an angle ever being measured.

Two jobs are handled here: scaling a triangle from one known side of its partner, and checking six measured sides to see whether two shapes really match.

The area is where intuition fails. Every length scales by the factor k, but area scales by k²: the tree's triangle has 44.4444 times the area of the post's, not 6.6667 times.

What is similar triangles calculator?

Two triangles are similar when one is a scaled copy of the other, possibly rotated or reflected. Their corresponding angles are equal and their corresponding sides share one ratio, the scale factor k. A k above 1 is an enlargement, below 1 a reduction, and exactly 1 means the triangles are congruent, identical in size as well as shape. Every length tied to the triangle, from altitudes and medians to the radii of its inscribed and circumscribed circles, scales by the same k.

How to use this calculator.

1. 01Choose Scale if you know all three sides of one triangle and one side of the other, or Check if you have all six lengths and want to know whether the shapes match.
2. 02Enter triangle 1's sides in a₁, b₁ and c₁ in any unit, keeping all six lengths in the same unit. Each side must be strictly shorter than the other two added together, or the triangle cannot exist.
3. 03Enter in a₂ the side of triangle 2 that matches a₁: the one opposite the same angle, which is not necessarily the shortest side.
4. 04In Check mode, enter b₂ and c₂ matched to b₁ and c₁ in the same way. In Scale mode those two boxes are ignored and calculated for you.
5. 05Read the scale factor, then the area ratio beside it, which is the scale factor squared when the triangles are similar.

The formula.

Scale mode reads the scale factor from the one known pair, k = a₂ ÷ a₁, and applies it to the other two sides: b₂ = k·b₁ and c₂ = k·c₁. The perimeter follows directly: P₂ = k·a₁ + k·b₁ + k·c₁ = k·P₁, so P₂ ÷ P₁ = k.

Both areas come from Heron's formula, A = √(s(s−a)(s−b)(s−c)), where s = (a + b + c) ÷ 2 is the semiperimeter. Multiplying every side by k multiplies s by k too, so each of the four factors under the root gains a factor of k, the product gains k⁴, and its square root gains k². That is why A₂ ÷ A₁ = k², the result Euclid gives as Book VI, Proposition 19, where similar triangles stand in the "duplicate ratio" of their sides. Because each area is calculated from its own side lengths, the k² relationship appears as a checkable result rather than being assumed.

Check mode forms the three ratios a₂/a₁, b₂/b₁ and c₂/c₁ and measures their relative spread, (largest − smallest) ÷ smallest. A spread of 10⁻⁹ or less counts as identical. Up to 0.005, half a percent, the triangles are reported as similar within measurement tolerance; anything wider is reported as not similar. The half-percent line is a practical convention chosen for measured lengths, not a mathematical standard.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Euclid, Elements, Book VI, Proposition 4 — 'In equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles.' David E. Joyce web edition of the Heath translation, Clark University; statement confirmed verbatim on retrieval 2026-07-29. Open access.
2. [2]Euclid, Elements, Book VI, Proposition 5 — the converse: 'If two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides.' This is what licenses an SSS-only similarity test. Joyce web edition; statement confirmed verbatim on retrieval 2026-07-29. Open access.
3. [3]Euclid, Elements, Book VI, Proposition 19 — 'Similar triangles are to one another in the duplicate ratio of the corresponding sides.' The independent authority used to check this page's area-ratio output: for k = 2.5 it requires A₂/A₁ = 6.25, and Heron's formula applied to the two triangles independently gives 37.5/6 = 6.25 — agreement. Joyce web edition; statement confirmed verbatim on retrieval 2026-07-29. Open access.
4. [4]Wolfram Research, MathWorld — 'Similar Triangles' (live revision retrieved 2026-07-30). Defines similar triangles as triangles whose triples of vertex angles are the same — the angle form of the definition. The entry is brief and does not itself enumerate the AA, SSS and SAS test forms; those are standard textbook restatements of Euclid's Book VI, Propositions 4 and 5, cited above. Open access.
5. [5]Wolfram Research, MathWorld — 'Heron's Formula' (live revision retrieved 2026-07-29). A = √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2, the formula this page uses to compute both triangle areas from side lengths alone. Open access.