How do you find the area of a triangle from three sides?

Use Heron's formula. Compute the semi-perimeter s = (a + b + c) / 2, then area = square root of s(s − a)(s − b)(s − c).

How do you find an angle from the three sides?

Use the law of cosines. For the angle opposite side a: cos(A) = (b² + c² − a²) / (2bc), then take the inverse cosine.

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Math & Statistics

Triangle Calculator

Solve a triangle from three sides, or from two sides and the included angle. Get every angle, the area, the perimeter, the type, and a scaled diagram.

Area

Right scalene triangle

Angles (A / B / C)—

Sides (a / b / c)—

Perimeter—

Triangle diagram

Drawn to scale from your inputs, with sides and angles labelled.

How it works

Given all three sides, the calculator finds each angle with the law of cosines and the area with Heron's formula. Given two sides and the angle between them, it first finds the third side, then solves the rest the same way.

Law of cosines: cos(A) = (b² + c² − a²) / (2bc)
Heron's area: s = (a+b+c)/2, Area = √( s(s−a)(s−b)(s−c) )
Third side (SAS): c = √( a² + b² − 2ab·cos C )

Worked example

A 6-8-10 triangle: the semi-perimeter is 12, so the area is √(12·6·4·2) = √576 = 24. Because 6² + 8² = 10², the largest angle is exactly 90°, making it a right triangle.

Frequently asked questions

How do you find the area from three sides?

Use Heron's formula: semi-perimeter s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)).

How do you find an angle from the sides?

The law of cosines: cos(A) = (b² + c² − a²) / (2bc), then take the inverse cosine.

What makes a valid triangle?

The triangle inequality: any two sides must add up to more than the third.