Triangle Calculator

How This Calculator Works

A triangle’s area, perimeter, and all three interior angles can be found from just its three side lengths, with no need to measure a height or angle directly. Enter the three sides (in whatever unit is convenient — the calculator converts internally) and it returns the area, perimeter, each angle, and two classifications: by side lengths (equilateral, isosceles, or scalene) and by angles (acute, right, or obtuse).

Not every three lengths can actually form a triangle. The “triangle inequality” requires each side to be shorter than the sum of the other two — three drinking straws of length 1, 2, and 10 simply can’t be arranged to meet at three corners, no matter how you angle them. This calculator checks that first and explains clearly if the lengths you entered don’t work.

The Formula

Area comes from Heron’s formula, using the semi-perimeter s (half the perimeter):

$$\text{Area} = \sqrt{\vD{s}(\vD{s} - \vA{a})(\vD{s} - \vB{b})(\vD{s} - \vC{c})}$$

Each angle comes from the law of cosines, solved for the angle opposite a given side — for example, the angle opposite side a:

$$\text{Angle A} = \cos^{-1}\left(\frac{\vB{b}^2 + \vC{c}^2 - \vA{a}^2}{2\vB{b}\vC{c}}\right)$$

The same formula, with the sides rotated, gives angles B and C. All three angles always add up to exactly 180°.

Worked Example

A triangle with sides 7, 8, and 9:

1. Semi-perimeter: $(7 + 8 + 9) \div 2 = 12$.
2. Area: $\sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} \approx 26.83$.
3. Angle opposite the 7-side: $\cos^{-1}\left(\frac{8^2 + 9^2 - 7^2}{2 \times 8 \times 9}\right) \approx 48.19°$.
4. Angle opposite the 8-side: $\approx 58.41°$.
5. Angle opposite the 9-side: $\approx 73.40°$ — the largest angle, but still under 90°.

All three sides are different lengths and every angle is under 90°, so this triangle is scalene and acute.