Triangle Calculator
Calculate triangle area, perimeter, semi-perimeter, and interior angles.
Which rule to reach for
What you can solve depends on what you are given. With three sides (SSS) or two sides and the included angle (SAS), use the law of cosines. With two angles and any side (AAS or ASA), use the law of sines. Angles always total 180°, so knowing two gives the third for free.
The awkward case is two sides and a non-included angle (SSA), known as the ambiguous case, because it can produce two valid triangles, one, or none. The law of sines will hand you an acute angle when the obtuse supplement may fit equally well, and a calculator's inverse sine cannot tell you which — check whether the supplementary angle still leaves the total under 180°, and if it does, both triangles are real.
For area, pick by what you know: ½ × base × height with a perpendicular height,½ab·sin(C) with two sides and the angle between them, and Heron's formula with three sides and no angles at all. One sanity check before any of this: the triangle inequality requires each pair of sides to sum to more than the third, or no triangle exists.
Frequently Asked Questions
How is triangle area calculated from three sides (SSS)?
Using Heron’s formula: Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (a+b+c)/2.
What is the sum of angles in any triangle?
The sum of interior angles in any Euclidean triangle is always 180 degrees (π radians).
How do I find the area of a triangle without the height?
Heron's formula uses the three sides: with s as half the perimeter, area = sqrt(s(s-a)(s-b)(s-c)). If two sides and the included angle are known, area = 1/2 x a x b x sin(C) is quicker.
When do I use the law of sines versus the law of cosines?
Law of sines (a/sin A = b/sin B) suits angle-side-angle and side-side-angle cases. Law of cosines (c^2 = a^2 + b^2 - 2ab cos C) handles side-side-side and side-angle-side, where the law of sines cannot start. Note the SSA case is ambiguous and can yield two valid triangles.