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Oblique Triangle Solver (Law of Cosines)

Solve any triangle — not just right triangles — for its remaining sides, angles, area, and perimeter, given three sides (SSS) or two sides and the included angle (SAS).

Side a
5
Side b
7
Side c
10
Angle A (opposite side a)
27.66°
Angle B (opposite side b)
40.54°
Angle C (opposite side c)
111.8°
Area
16.2481
Perimeter
22

How it works

This site's Right Triangle Calculator covers only triangles with a 90° angle. Most triangles aren't right triangles, so solving them needs the Law of Cosines instead: c² = a² + b² − 2ab·cos(C), where C is the angle between sides a and b.

Given three sides (SSS), the Law of Cosines is rearranged to solve for each angle directly — e.g. angle A = cos⁻¹((b² + c² − a²) ÷ 2bc) — after checking the three sides can actually form a triangle (each side must be shorter than the sum of the other two). Given two sides and their included angle (SAS), the same formula solves for the missing side first, then the remaining angles.

Area is ½·a·b·sin(C), which is equivalent to Heron's formula but only needs the angle between two sides rather than all three side lengths — since C is already known by the time area is calculated in either mode, this is simpler to compute directly. When C happens to be 90°, every result here matches the Right Triangle Calculator exactly, since a right triangle is just a special case of this general one.

FAQ

Why can't I just enter any three numbers as sides?

Three lengths only form a triangle if each one is shorter than the sum of the other two (the triangle inequality) — otherwise the two shorter sides simply can't reach each other to close the shape. This calculator checks that before solving and reports an error if the sides you entered can't form a real triangle.

What's the difference between this and the Right Triangle Calculator?

The Right Triangle Calculator assumes one angle is exactly 90° and uses the simpler Pythagorean theorem, which only applies to that special case. This calculator handles any triangle — acute or obtuse — using the more general Law of Cosines, at the cost of needing three known measurements (three sides, or two sides and the angle between them) instead of two.

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