Triangle Solver

What this tool is used for

• Solving a triangle from three known values in a geometry problem.
• Finding a missing side for a construction or a layout.
• Checking a homework answer with the working shown.
• Solving a surveying or a navigation triangle.
• Confirming that three given values actually determine a triangle.

Frequently Asked Questions

What does it take to determine a triangle?

Three pieces of information including at least one side: SSS, SAS, ASA, AAS or SSA. AAA fixes the shape but not the size, giving infinitely many similar triangles — which is why three angles alone can never be solved.

What is the ambiguous SSA case?

Given two sides and a non-included angle, there can be two valid triangles, one, or none — because the sine rule yields an angle whose supplement is equally valid. This is the one configuration where a solver must return two answers.

When do I use the sine rule versus the cosine rule?

Sine rule (a/sin A = b/sin B) when you have an angle paired with its opposite side. Cosine rule (c² = a² + b² − 2ab·cos C) for SSS and SAS, where no such pair exists.

Which side lengths cannot form a triangle?

Any where the two shorter sides do not exceed the longest — the triangle inequality. 1, 2, 3 fails because 1 + 2 equals 3 exactly, which degenerates into a straight line rather than a triangle.

How is the area found without a height?

Heron's formula from the three sides, or ½ab·sin C from two sides and their included angle. Heron's is numerically unstable for very thin triangles, where a rearranged version is used instead.

Why does knowing all three angles not determine a triangle?

Because the angles fix only the shape, not the size — every similar triangle satisfies them. At least one side length is needed to pin down the scale, which is why AAA is the one combination that has no unique solution.

What are the circumradius and inradius?

The radii of the circle through all three vertices and of the circle touching all three sides. They fall out of the sine rule and the area respectively, and are what you need for anything inscribed in or circumscribed about the triangle.

Common errors and gotchas

• Supplying values that describe no triangle, such as sides that violate the triangle inequality.
• Hitting the ambiguous SSA case, where two different triangles satisfy the same three values.
• Mixing degrees and radians, which is the classic trigonometry error.
• Assuming angles sum to 180 in a non-Euclidean context, such as on a sphere.
• Rounding intermediate values, which compounds through the law of sines.