Triangle Calculator
Every side, angle and the area of a triangle from the three values you know.
Good to know. Three sides fix a triangle completely, which is why a triangle frame doesn’t wobble and a square one does.
How it works
In plain words
With three sides, Heron’s formula gives the area and the law of cosines gives each angle. Two sides and the angle between them give the third side by the law of cosines; two angles and a side give the others by the law of sines. The angles always add up to 180°.
The formula
c² = a² + b² − 2ab·cos C a ÷ sin A = b ÷ sin B = c ÷ sin C area = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2
Worked example
The sides 3, 4 and 5 give s = 6 and an area of √(6 × 3 × 2 × 1) = 6. The angles are 36.87°, 53.13° and exactly 90°: the classic right triangle.
Sources
No outside data or rates: this is plain arithmetic.
5 automated tests check its answers on every change.
About the triangle calculator
A triangle has six measurements, three sides and three angles, and three of them are almost always enough to find the rest, as long as at least one is a side. This calculator handles the common cases: three sides, two sides with the angle between them, two angles with the side between them, and a right triangle from its two shorter sides.
Sides are named a, b and c, and each angle takes the capital letter of the side opposite it: angle A faces side a. The law of cosines, a version of Pythagoras that works for any triangle, links the three sides and one angle. The law of sines says each side divided by the sine of its opposite angle gives the same number.
Not every three numbers make a triangle. Each side has to be shorter than the other two put together, or the ends can’t meet; and two angles must leave something over from 180°. The calculator says when that happens.
The units are whatever you use for the sides: the area comes out in the same unit squared. For a right triangle’s missing side, the Pythagorean theorem calculator is quicker; for slopes, the slope calculator.
Questions people ask
How do I find the area of a triangle from three sides?
With Heron’s formula: s = half the perimeter, area = √(s(s − a)(s − b)(s − c)). Sides 3, 4 and 5 give 6.
How do I find a missing angle?
If you know two angles, it is 180° minus both. With three sides, use the law of cosines, as this calculator does.
Can any three lengths make a triangle?
No. Each must be shorter than the other two added together: 2, 3 and 6 can’t meet.