Triangle Solver

Enter any 3 values — at least one side. MathLive finds the rest, step by step.

Sides

Side a

Side b

Side c

Angles (angle A is opposite side a)

Angle A (°)

Angle B (°)

Angle C (°)

Angles in Decimals

Tip: you can type values like 3/4, sqrt(2) or pi/3 (radians). Press Enter to solve.

Examples

Your Triangle

Results

Your results will appear here.

Step-by-Step Solution

How the Triangle Solver Works

A triangle has three sides (a, b, c) and three angles (A, B, C). Each side is named after the angle opposite it, so side a faces angle A. Knowing any three of these six values, including at least one side, is usually enough to find all the others. MathLive works out which case you entered, applies the right rule, shows every step, and draws the triangle to scale.

The two rules behind it

Law of Cosines: c² = a² + b² − 2ab·cos C. It is used when you know three sides (SSS) or two sides and the angle between them (SAS). Law of Sines: a / sin A = b / sin B = c / sin C. It is used when you know two angles and a side (ASA or AAS), or two sides and an angle that is not between them (SSA). The three angles always add up to 180°.

Worked example

For sides a = 3, b = 4, c = 5: cos C = (3² + 4² − 5²)/(2·3·4) = 0/24 = 0, so C = 90° — a right triangle. Then cos A = (4² + 5² − 3²)/(2·4·5) = 32/40 = 0.8, so A ≈ 36.87°, and B = 180° − 90° − 36.87° = 53.13°. The area is ½·3·4 = 6.

Frequently asked questions

Why can there be two triangles?

With two sides and an angle that is not between them (SSA), the side opposite the angle can sometimes swing to two different positions. For a = 7, b = 10 and A = 40°, sin B = 10·sin 40° / 7 ≈ 0.918, so B ≈ 66.7° or B ≈ 113.3°. Both give valid triangles, and the solver draws each one. If the side is too short to reach, there is no triangle at all.

Can I enter three angles?

Three angles fix the shape but not the size — every enlarged copy has the same angles — so at least one side is needed.

How is the area calculated?

Area = ½·b·c·sin A, using two sides and the angle between them. The solver also lists the perimeter, the three heights, the inradius r and the circumradius R (see the pictures below).

Degrees or radians?

Choose the unit from the menu. Switching converts any angles you already typed, so your triangle stays the same. For a right triangle you can also explore its angles on the Interactive Unit Circle.

Inradius and circumradius

Every triangle has two special circles. The solver gives the radius of each one.

Inradius r — the radius of the largest circle that fits inside the triangle, touching all three sides (the incircle). Its center I is where the three angle bisectors meet. r = Area ÷ s, where s = (a + b + c) ÷ 2

Circumradius R — the radius of the circle that passes through all three corners A, B and C (the circumcircle). Its center O is where the perpendicular bisectors of the sides meet. R = a ÷ (2·sin A) = b ÷ (2·sin B) = c ÷ (2·sin C)

Example (3-4-5 triangle): s = 6 and Area = 6, so r = 6 ÷ 6 = 1. The longest side is 5 and it faces the 90° angle, so R = 5 ÷ (2·sin 90°) = 2.5. In every right triangle the circumcenter is the middle of the longest side (the hypotenuse), so R is always half of it.