Triangle Solver Calculator

Three of the four cases this calculator handles come straight from how triangles are classically solved. One is two sides and the included angle, called SAS. Another is a side and the two angles adjacent to it, called ASA. The third is a side, the angle opposite it, and an angle adjacent to it, called AAS ⁴.

The fourth case is three known sides, and it belongs to the law of cosines instead. That law is useful for solving a triangle when all three sides, or two sides and their included angle, are given. That is why the calculator reaches for it in both the SSS case and the SAS case ¹.

The sum of the measures of the interior angles of a triangle is always 180 degrees. That single fact allows the calculator to find a third angle whenever you give it two others. It is why the ASA and AAS cases only need two angles typed in, not three ⁵.

The triangle inequality states that the sum of the lengths of any two sides must be greater than or equal to the length of the third side. A triangle with three given positive side lengths exists only if those lengths satisfy that inequality. Check this before trusting an SSS result built from a real-world measurement ⁵.

The law of cosines relates the lengths of a triangle's sides to the cosine of one of its angles. It ties together three companion equations, one for each side: c² = a² + b² − 2ab cos γ, and the same pattern for a and b ¹.

When you give two sides and the angle between them, the calculator finds the missing side with c = √(a² + b² − 2ab cos γ). When you give three sides instead, it finds each angle with γ = arccos((a² + b² − c²) / (2ab)) ¹.

The law of sines is an equation relating the lengths of a triangle's sides to the sines of its angles. It holds for every triangle, not just special ones ².

That relationship computes the remaining sides of a triangle once two angles and a side are known. Wikipedia calls this technique triangulation. It is exactly what the ASA and AAS cases give the calculator to work with ².

The law of sines is written a / sin α = b / sin β = c / sin γ. Each lowercase side letter is paired with the Greek angle letter directly across from it. Side a sits opposite angle α, side b sits opposite angle β, and side c sits opposite angle γ. That pairing is exactly how this calculator labels its own inputs and outputs ².

Heron's formula gives the area of a triangle from its three side lengths alone, with no angle needed. As long as those lengths obey the strict triangle inequality, they define a real triangle whose area is a positive number ³.

The formula sets s = ½(a + b + c), then A = √(s(s−a)(s−b)(s−c)). For sides of 4 m, 13 m, and 15 m, half the perimeter is ½(4 + 13 + 15) = 16. The area then works out to √(16 · 12 · 3 · 1) = 24 ³.

To find an unknown angle, the law of cosines is safer than the law of sines. The sine of an angle does not uniquely determine that angle. If sin β = 0.5, β could be 30° or 150°, and the sine alone cannot tell you which. That is why this calculator's SSS and SAS cases lean on the law of cosines for angles ⁴.

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles. Because of that, this calculator needs no separate right-triangle method; it solves a right triangle the same way it solves any other one ¹.

These sources describe how a triangle's sides and angles relate through the law of cosines, the law of sines, and Heron's formula. They say nothing about a right triangle's lines of symmetry, about triangles with equal sides, or about whether right triangles are similar to one another. They also do not describe a median, an inradius, or a circumradius, so this page does not calculate any of those ¹.

Distance measurement by triangulation is one practical use of triangle solving that these sources point to directly ⁴.

All types of triangles are commonly found in real life. Isosceles triangles show up in gables and pediments, and the equilateral triangle appears in the yield sign ⁵.

You may also be wondering about medians, inradius, and circumradius, how many lines of symmetry a right triangle has, whether a right triangle can have equal sides, or whether all right triangles are similar. This page won't guess past what's actually said ⁵.