Calculate the area of any triangle from its three side lengths using Heron’s formula, or solve for a missing side when you know the area and two sides.
Heron's Formula
Heron's formula finds the area of a triangle from the lengths of its three sides, without needing the height or any angles. First find the semi-perimeter, then plug it into the area equation.
s = (a + b + c) / 2
A = sqrt(s(s - a)(s - b)(s - c))
- A = area of the triangle (square units)
- s = semi-perimeter, half of the triangle's total perimeter
- a, b, c = the lengths of the three sides
Enter the three side lengths and the calculator computes the semi-perimeter, then the area. When you switch the "I want to solve for" selector to a side, you instead supply the area and the other two sides, and the calculator works the area equation backward to find the missing side. The decimal-places option controls rounding, and the advanced toggle shows the semi-perimeter and the worked steps so you can check the result by hand. The three sides must satisfy the triangle inequality (each side shorter than the sum of the other two) or no real triangle exists.
Triangle Area for Common Side Sets
These values come straight from Heron's formula and are useful for a quick sanity check against your own inputs.
| Sides a, b, c | Semi-perimeter s | Area A |
|---|---|---|
| 3, 4, 5 | 6 | 6 |
| 5, 5, 5 | 7.5 | 10.83 |
| 6, 8, 10 | 12 | 24 |
| 7, 8, 9 | 12 | 26.83 |
| 13, 14, 15 | 21 | 84 |
When the sides do not form a triangle
| Sides a, b, c | Result |
|---|---|
| 2, 2, 5 | Invalid: 2 + 2 is less than 5 |
| 1, 1, 2 | Degenerate: area is 0, sides lie on a line |
Example Problems
Example 1. A triangle has sides of 7, 8, and 9.
- Semi-perimeter: s = (7 + 8 + 9) / 2 = 12
- Area: A = sqrt(12(12 - 7)(12 - 8)(12 - 9)) = sqrt(12 * 5 * 4 * 3) = sqrt(720)
- A = 26.83 square units
Example 2. A triangle has sides of 6, 8, and 10.
- Semi-perimeter: s = (6 + 8 + 10) / 2 = 12
- Area: A = sqrt(12(12 - 6)(12 - 8)(12 - 10)) = sqrt(12 * 6 * 4 * 2) = sqrt(576)
- A = 24 square units (this is a right triangle, so you can confirm with 1/2 * 6 * 8 = 24)
FAQ
What do you need to use Heron's formula?
You only need the lengths of all three sides. Unlike the base-times-height method, Heron's formula does not require the height or any angle, which makes it useful when you know the sides but cannot measure the height directly.
What if the calculator says the triangle is invalid?
The three sides must satisfy the triangle inequality: each side must be shorter than the sum of the other two. If one side is equal to or longer than the other two added together, no real triangle exists and the area is either zero or undefined.
Can Heron's formula find a missing side?
Yes. If you know the area and two of the sides, you can rearrange the area equation to solve for the third side. Switch the solve-for selector to the side you want, enter the area and the two known sides, and the calculator returns the missing length.
