Triangle Solver

Enter any valid combination of sides and angles to solve a triangle. The solver uses the Law of Sines and Law of Cosines to find all missing values, with step-by-step solutions shown in full. All calculations run in your browser.

Three sides

Side a

Side b

Side c

Side a

Side b

Side c

Angle A

38.2132°

Angle B

60°

Angle C

81.7868°

Area

17.3205

Perimeter

Step-by-Step Solution

Reference Guide

Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem to any triangle. It relates the three sides of a triangle to the cosine of one of its angles.

General form

c^2 = a^2 + b^2 - 2ab\cos C

Use this law when you know all three sides (SSS) or two sides and the included angle (SAS). For SSS, rearrange to find each angle:

Solving for an angle

C = \cos^{-1}\!\left(\frac{a^2 + b^2 - c^2}{2ab}\right)

Law of Sines

The Law of Sines states that the ratio of each side to the sine of its opposite angle is the same for all three pairs.

General form

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Use this law for ASA (two angles and the included side), AAS (two angles and a non-included side), and SSA (two sides and a non-included angle). For ASA and AAS, find the third angle first since angles sum to 180 degrees.

Triangle Area

Several formulas compute the area of a triangle depending on what information is available.

Two sides and included angle

A = \tfrac{1}{2}\,a\,b\,\sin C

Heron's formula (three sides known)

A = \sqrt{s(s-a)(s-b)(s-c)}

In Heron's formula, $s = \tfrac{1}{2}(a+b+c)$ is the semi-perimeter. This formula is useful when you know all three sides but no angles.

The Ambiguous Case (SSA)

When two sides and a non-included angle are given, the triangle may not be unique. Depending on the measurements, there can be zero, one, or two valid triangles.

Given sides $a$ and $b$ with angle $A$ opposite side $a$ :

No triangle if $a < b\sin A$ . The side is too short to reach the opposite side.
One triangle if $a = b\sin A$ (a right triangle) or if $a \ge b$ .
Two triangles if $b\sin A < a < b$ . The Law of Sines gives two possible values for the unknown angle.