Similar Triangles Calculator

Background.

The Quanta similar triangles calculator does the two jobs this topic actually generates. Give it all three sides of one triangle plus a single corresponding side of a second, and it returns the scale factor, the two missing sides, both areas, the perimeter ratio and the area ratio. Give it all six sides instead and it tells you whether the two triangles are similar at all, and by how much the three side ratios disagree if they are not. With the default 3-4-5 triangle scaled from a₂ = 7.5, the scale factor comes back as 2.5, the missing sides as 10 and 12.5, the perimeter ratio as 2.5 and the area ratio as 6.25.

Two triangles are similar when they have the same shape at possibly different sizes: equal angles, proportional sides. Euclid proved that those two descriptions are the same condition. Book VI, Proposition 4 of the Elements says equiangular triangles have proportional sides; Proposition 5 is the converse, that proportional sides force equal angles. That equivalence is what makes this calculator possible without ever measuring an angle — six lengths are enough to settle the question.

The part people get wrong is the area. Doubling every side of a triangle does not double its area, it quadruples it. Scale by 2.5 and the area goes up by 6.25. Euclid states it as Book VI, Proposition 19: "Similar triangles are to one another in the duplicate ratio of the corresponding sides" — duplicate ratio being the classical phrase for the squared ratio. This page computes both areas independently from the side lengths using Heron's formula rather than multiplying one by k², so the k² relationship shows up as a result you can check rather than an assumption you have to accept. It is also the reason the area ratio is worth reading in check mode: when two triangles are not similar, the area ratio will visibly fail to be the square of the scale factor.

One caveat belongs beside the answer rather than three screens below it: the correspondence is your job, not the calculator's. It pairs a₁ with a₂, b₁ with b₂ and c₁ with c₂ strictly in the order you type them. Two triangles that genuinely are similar will still be reported as not similar if you enter the sides of one in ascending order and the sides of the other in descending order. If you get an unexpected verdict, re-order the second triangle's sides before concluding the shapes differ — the verdict text prints the three ratios so you can see which pairing is out of line.

The check-mode verdict uses a tolerance, and it is this page's own convention rather than a standard. Ratios that agree to within one part in a billion are called identical. Ratios that agree to within half a percent are reported as similar within measurement tolerance, because someone holding a tape measure cannot produce ratios that match to fifteen digits and a flat "not similar" would be useless to them. Anything wider is reported as not similar, with the spread named as a percentage. Where you draw that line for your own work is a judgement about your instrument, not about the geometry.

Below the calculator you will find the derivation of both the linear perimeter scaling and the quadratic area scaling, a fully worked 3-4-5 example you can verify by hand in under a minute, the three standard similarity criteria (AA, SSS and SAS) and which of them this page implements, and the primary sources: Euclid's Elements Books VI.4, VI.5 and VI.19, and MathWorld's entries for similar triangles and Heron's formula.

What is similar triangles calculator?

Two triangles are similar when one is a scaled copy of the other — same shape, possibly different size, possibly reflected or rotated. Formally, triangles ABC and DEF are similar when their corresponding angles are equal (A = D, B = E, C = F) and, equivalently, when their corresponding sides are in a constant ratio (DE/AB = EF/BC = FD/CA). Euclid proved the equivalence in two adjacent propositions of Book VI of the Elements: Proposition 4 states that 'in equiangular triangles the sides about the equal angles are proportional where the corresponding sides are opposite the equal angles', and Proposition 5 states the converse, that 'if two triangles have their sides proportional, then the triangles are equiangular with the equal angles opposite the corresponding sides'. That constant ratio is called the scale factor and is usually written k. Three criteria are enough to establish similarity in practice, and every geometry course teaches the same three. AA: if two angles of one triangle equal two angles of another, the triangles are similar — the third angle follows automatically from the 180° angle sum, which is why AAA is redundant. SSS: if all three pairs of corresponding sides are in the same ratio, the triangles are similar. SAS: if two pairs of corresponding sides are in the same ratio and the angles between those pairs are equal, the triangles are similar. This calculator implements the SSS criterion, because sides are what a tape measure gives you and angles are not. Similarity is strictly weaker than congruence: congruent triangles are similar with k = 1, but similar triangles need not be the same size. Similarity also ignores reflection — a triangle and its mirror image have identical side lengths and are similar under every criterion above, so no side-based test can tell them apart.

How to use this calculator.

1. 01Pick the mode. Choose Scale when you know all three sides of one triangle and exactly one side of the other — that is the standard homework and field-measurement case. Choose Check when you have measured all six lengths and want to know whether the shapes actually match.
2. 02Enter triangle 1's three sides in a₁, b₁ and c₁, in any unit you like, as long as all six numbers use the same one. The three lengths must satisfy the triangle inequality: each side strictly shorter than the other two added together.
3. 03Enter the side of triangle 2 that corresponds to a₁ in the a₂ field. Correspondence matters more than order of size: a₂ must be the side sitting opposite the same angle as a₁, not simply the shortest side of the second triangle.
4. 04In check mode, fill in b₂ and c₂ as well, matching b₁ and c₁ in the same way. In scale mode those two fields are ignored — the calculator computes them from the scale factor.
5. 05Read the scale factor at the top. Everything in triangle 2 that is a length — sides, perimeter, height, median, inradius, circumradius — is that many times the matching length in triangle 1.
6. 06Read the area ratio and note that it is the square of the scale factor, not the scale factor. This is the single most common error in the topic and the reason the two ratios are shown side by side.
7. 07If check mode reports 'not similar', look at the three ratios printed in the verdict before you accept it. Two ratios agreeing and one disagreeing usually means the sides of the second triangle were entered in the wrong correspondence, not that the triangles differ.

The formula.

Similarity means one constant of proportionality governs every length. Write it as k. Then a₂ = k·a₁, b₂ = k·b₁ and c₂ = k·c₁, and scale mode is nothing more than reading k off the one pair you know and applying it to the other two.

PERIMETER. P₂ = a₂ + b₂ + c₂ = k·a₁ + k·b₁ + k·c₁ = k(a₁ + b₁ + c₁) = k·P₁, so P₂ ÷ P₁ = k. Perimeter is a length, and every length scales by the same factor. The same argument works for any other length attached to the triangle — an altitude, a median, an angle bisector, the radius of the inscribed circle, the radius of the circumscribed circle. All of them scale by k and none of them by anything else.

AREA. Area does not scale by k. Take Heron's formula, A = √(s(s−a)(s−b)(s−c)) with s = (a+b+c) ÷ 2. Scaling every side by k scales s by k as well, so each of the four factors inside the square root picks up one power of k, the radicand picks up k⁴, and the square root picks up k². Therefore A₂ ÷ A₁ = k². Euclid reaches the same conclusion geometrically in Elements Book VI, Proposition 19: 'Similar triangles are to one another in the duplicate ratio of the corresponding sides.' The intuition is that area is a product of two lengths, so it carries two factors of k, in the same way that a volume would carry three.

WORKING THE DEFAULT ALL THE WAY THROUGH. Triangle 1 is 3, 4, 5 and the known side of triangle 2 is a₂ = 7.5. The scale factor is k = 7.5 ÷ 3 = 2.5. Then b₂ = 4 × 2.5 = 10 and c₂ = 5 × 2.5 = 12.5. Perimeters: P₁ = 3 + 4 + 5 = 12 and P₂ = 7.5 + 10 + 12.5 = 30, so P₂ ÷ P₁ = 30 ÷ 12 = 2.5, which is k as promised. Areas by Heron: for triangle 1, s = 12 ÷ 2 = 6 and A₁ = √(6 × 3 × 2 × 1) = √36 = 6; for triangle 2, s = 30 ÷ 2 = 15 and A₂ = √(15 × 7.5 × 5 × 2.5) = √1406.25 = 37.5. The area ratio is 37.5 ÷ 6 = 6.25, which is 2.5², confirming Euclid VI.19 against numbers computed without ever using k.

CHECK MODE AND ITS CLASSIFIER. With all six sides supplied, the calculator forms the three ratios r_a = a₂ ÷ a₁, r_b = b₂ ÷ b₁ and r_c = c₂ ÷ c₁, and measures their relative spread, (max r − min r) ÷ min r. That spread is the scalar the verdict is based on, and it is classified unrounded at 40 significant digits — the percentage printed in the verdict is a display of it, never the value the decision is made from. A spread of 1e-9 or less is called identical. A spread up to 0.005 (0.5%) is reported as similar within measurement tolerance. Anything wider is reported as not similar. Both edges are inclusive on the tighter side and both are covered by tests that hit them exactly.

ROUNDING STAGE. All arithmetic runs at 40 significant digits in Decimal, and the scale factor is not rounded before b₂ and c₂ are derived from it — rounding k first would put visible error into the area, which is quadratic in k. Rounding happens once, at the return boundary, to 10 decimal places.

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.