Triangle Calculator
Calculate a triangle's angles, area and perimeter from the lengths of its three sides using the law of cosines and Heron's formula.
- Angle A44.42° · 25%
- Angle B57.12° · 32%
- Angle C78.46° · 44%
Every triangle's three angles add up to exactly 180° — this donut shows how that total splits between the three corners.
- Perimeter18
- Angle A (opp. side a)44.42°
- Angle B (opp. side b)57.12°
- Angle C (opp. side c)78.46°
ƒShow your work
Area = √(s(s−a)(s−b)(s−c)), s = perimeter/2
About the Triangle Calculator
Given just the three side lengths of a triangle, this calculator works out everything else — its area, perimeter, and all three interior angles — without needing a height or an angle measurement to start from. That's a common situation in geometry homework, but also in practical measuring: a carpenter or landscaper who's measured three physical sides and needs the angles or area without setting up a protractor.
It uses two classical results that between them solve any triangle from its sides alone: Heron's formula for area, and the law of cosines for the angles.
How it’s calculated
Heron's formula finds area from the three sides directly: first compute the semi-perimeter s = (a+b+c)/2, then Area = √(s(s−a)(s−b)(s−c)). It works for any triangle, right or not, without ever needing to know a height.
The law of cosines finds each angle from the three sides: for the angle opposite side a, cos(A) = (b² + c² − a²) / (2bc), and then A itself is the inverse cosine (arccos) of that value. Running this once for each side gives all three angles, and they'll always sum to exactly 180°.
Not every three lengths can actually form a triangle — the triangle inequality requires that any two sides added together must be longer than the third side. If that fails, no triangle exists with those measurements, which is why this calculator checks it before computing anything.
Frequently asked questions
How do you find a triangle's area from just its three sides?
Use Heron's formula: compute the semi-perimeter (half the total perimeter), then take the square root of the semi-perimeter times its difference from each of the three sides. No height measurement is needed.
What makes three lengths a valid triangle?
The triangle inequality: the sum of any two sides must be strictly greater than the third side. If a side is longer than or equal to the sum of the other two, the three lengths can't close into a triangle at all — they'd lie flat or not reach each other.
What's the difference between the law of cosines and the law of sines?
The law of cosines relates all three sides to one angle, and works when you know three sides (like here) or two sides and the included angle. The law of sines relates sides to the sines of their opposite angles and is used instead when you know two angles and a side, or two sides and a non-included angle.
Do a triangle's angles always add up to 180 degrees?
Yes, for any triangle on a flat (Euclidean) plane, the three interior angles always sum to exactly 180° — this is a basic geometric fact used here as a check: this calculator finds two angles directly and gets the third by subtracting from 180° rather than recomputing it.
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