📐Triangle Solver

About this tool

Enter any three of the six values, with at least one side. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The solver finds the missing sides and angles, then gives the area, perimeter, all three heights, the inradius and circumradius, the type of triangle and a scaled drawing. When two different triangles fit, both are shown.

How triangles are solved

Three sides fix a triangle, and so do two sides with the angle between them, or two angles with any side. The solver uses the law of cosines, c² = a² + b² − 2ab·cos C, to find a side opposite a known angle or an angle from three sides. With sides 8 and 6 and a 60° angle between them, the third side is about 7.211, and the angles are about 73.9°, 46.1° and 60°. The law of sines, a ÷ sin A = b ÷ sin B = c ÷ sin C, scales the sides when angles are known, using the fact that angles sum to 180°. Two sides and an angle that is not between them (SSA) is the ambiguous case. With a = 6, b = 8 and A = 40°, two different triangles fit, with B about 59° or about 121°, so both are shown. With a much shorter a, none fits. Area uses Heron's formula from the three sides, each height is 2 × area ÷ its base, the inradius is area ÷ half the perimeter, and the circumradius is a ÷ (2 sin A). Three angles alone fix only the shape, not the size, so at least one side is required. The Pythagorean theorem calculator handles right triangles, the area calculator covers other shapes and the scientific calculator handles trigonometry. Fractions, decimals and percentages: the same number three ways, and how to convert is a useful refresher.

Frequently asked questions