How to use this calculator
- Choose what you know.
- Enter the values. Side a is opposite angle A, b opposite B, and c opposite C.
- Sides can be in any unit; the area is in that unit squared.
Formula
The law of cosines finds angles from sides (or a side from two sides and an angle), the law of sines finds sides from angles, and Heron's formula gives the area from the three sides.
Law of cosines: c² = a² + b² − 2ab cos C
Law of sines: a / sin A = b / sin B = c / sin C
Heron: Area = √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2
Example (3-4-5)
- a = 3, b = 4, c = 5; s = 6.
- Area = √(6 × 3 × 2 × 1) = 6.
- C = arccos((9 + 16 − 25) / 24) = 90°, so it's a right triangle.
Frequently asked questions
Why does it say the sides don't form a triangle?
Any two sides must add up to more than the third (the triangle inequality). For example, 1, 2 and 5 can't form a triangle.
What about the ambiguous SSA case?
Two sides and a non-included angle can give zero, one or two triangles, so it isn't offered. Use SAS or SSS if you know the other values.
What are the inradius and circumradius?
The inradius is the radius of the largest circle that fits inside the triangle; the circumradius is the radius of the circle passing through all three vertices.