Heron's Formula
Calculate the area of any triangle from its three side lengths using Heron's formula: A = √(s(s-a)(s-b)(s-c)).
No angles needed.
The Formula
Step 1: s = (a + b + c) / 2
Step 2: A = √(s × (s - a) × (s - b) × (s - c))
Heron's formula calculates the area of a triangle when you know all three side lengths.
No angles or heights needed — just the three sides.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the triangle |
| a, b, c | The three sides of the triangle |
| s | The semi-perimeter (half the perimeter) |
Example 1
Find the area of a triangle with sides 7, 8, and 9
s = (7 + 8 + 9) / 2 = 24 / 2 = 12
A = √(12 × (12 - 7) × (12 - 8) × (12 - 9))
A = √(12 × 5 × 4 × 3) = √720
A ≈ 26.83 square units
Example 2
Find the area of a triangle with sides 13, 14, and 15
s = (13 + 14 + 15) / 2 = 42 / 2 = 21
A = √(21 × (21 - 13) × (21 - 14) × (21 - 15))
A = √(21 × 8 × 7 × 6) = √7,056
A = 84 square units
When to Use It
Use Heron's formula when:
- You know all three side lengths but no angles or heights
- Measuring a triangular plot of land where sides are easier to measure than heights
- The triangle is not a right triangle
- You want an exact area without needing to calculate any angles first