Triangle Congruence Calculator

Background.

Three side lengths settle everything about a triangle. That is the side-side-side (SSS) congruence rule: two triangles whose sides match pair by pair are congruent, identical in every angle and in area, however they are turned or flipped. It is also why three bars bolted at the corners cannot be pushed out of shape while a square frame folds into a rhombus, and why roof trusses and bridge girders are built from triangles.

This page puts the rule to work. Give it three lengths and it checks that they can close into a triangle, then returns the two quantities the sides alone decide: the perimeter, which is their sum, and the area, from Heron's formula. Heron's method needs no angles and no height, only the semiperimeter s, half the perimeter, and the three differences s − a, s − b and s − c.

The closure test is the triangle inequality. Each side must be shorter than the other two put together. If one side exactly equals the sum of the others, the shape collapses flat with zero area; if it is longer, the ends cannot meet. Both cases are refused.

The page does not compare two triangles or name a congruence rule, so treat its output as evidence rather than proof. Two triangles with different areas or perimeters are certainly not congruent. Matching results, though, prove nothing: sides of 5, 5, 6 and 5, 5, 8 both enclose an area of 12, and some different triangles share both area and perimeter. Congruence needs all three sides to match, or another recognised combination such as two sides and the angle between them.

Builders rely on the same fact when they set out a square corner with a 3-4-5 triangle: those three lengths, the default example here, close only into a right angle.

What is triangle congruence calculator?

Two triangles are congruent when they are exact copies: the same three side lengths and the same three angles, so one can be moved, rotated or reflected to sit perfectly on the other. The side-side-side criterion says that matching the three sides is enough on its own. The sides you enter define one such triangle, and the page reports the measurements every congruent copy shares: the perimeter, which is the sum of the sides, and the area, from Heron's formula.

How to use this calculator.

1. 01Enter the three side lengths in any order, all in the same unit. The perimeter comes out in that unit and the area in its square, so centimetres give square centimetres.
2. 02If the sides are refused, look at the longest one: it must be strictly shorter than the other two added together.
3. 03To test whether two triangles could be congruent, run each set of sides and compare. A different perimeter or area rules congruence out; for proof, check that the three sides match pair by pair.
4. 04For an irregular plot or floor, divide it into triangles, measure each triangle's three sides with a tape, and add the areas.

The formula.

With sides a, b and c, the semiperimeter is s = (a + b + c) ÷ 2, Heron's formula gives the area A = √[s(s − a)(s − b)(s − c)], and the perimeter is 2s. Each factor such as s − a is half the amount by which the other two sides exceed a, which is where the triangle inequality comes in: if any side reaches the sum of the other two, its factor drops to zero or below and there is no area to take a square root of. The page requires every factor to be strictly positive. Scaling all three sides by k multiplies the perimeter by k and the area by k². Area is most sensitive in flat triangles, where one factor is close to zero: sides of 3, 4 and 6.9 enclose only about 2.01 square units even though the perimeter is 13.9.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.