How a triangle is solved
Side a sits opposite angle A, side b opposite angle B and side c opposite angle C.
Three sides: a = 5, b = 6, c = 7. The law of cosines gives A = 44.42°, B = 57.12° and C = 78.46°. The area is 14.6969 (Heron's formula agrees: √(9 × 4 × 3 × 2) = √216) and the perimeter is 18. It's an acute scalene triangle.
Two sides and the angle between them: a = 5, b = 7, C = 60°. Then c = √(25 + 49 − 70 × 0.5) = √39 = 6.245, and the area is ½ × 5 × 7 × sin 60° = 15.1554.
Two angles and the side between them: A = 40°, B = 60°, c = 10. C = 80°, and the law of sines gives a = 10 × sin 40° ÷ sin 80° = 6.527 and b = 8.794.
Types of triangles
By sides, a triangle is equilateral (all three equal), isosceles (two equal) or scalene (none equal). By angles, it's acute (all under 90°), right (one exactly 90°) or obtuse (one over 90°). For right triangles, the Pythagorean theorem calculator is quicker.
Two sides and an angle that isn't between them (SSA) can give two different triangles, one or none, so that case is left out here to avoid a misleading single answer.
Frequently asked questions
How do I find a missing side of a triangle?
With two sides and the angle between them, use the law of cosines: c² = a² + b² − 2ab cos C. With two angles and a side, find the third angle and use the law of sines.
How do I find the angles from three sides?
Use the law of cosines rearranged: cos A = (b² + c² − a²) ÷ 2bc, then do the same for B, and C = 180° − A − B.
What is Heron's formula?
A way to find the area from three sides: with s = half the perimeter, area = √(s(s − a)(s − b)(s − c)).
Was this calculator helpful?