Heron's Formula
Calculate the area of any triangle from its three side lengths using Heron's formula.
Includes step-by-step examples.
The Formula
where s = (a + b + c) / 2
Heron's formula calculates the area of any triangle when you know all three side lengths. You do not need the height or any angles.
The formula is attributed to Heron of Alexandria (circa 10–70 CE), a Greek mathematician and engineer.
Variables
| Symbol | Meaning |
|---|---|
| A | Area of the triangle |
| a, b, c | The three side lengths |
| s | Semi-perimeter (half the perimeter) |
Example 1
A triangle has sides of 7, 8, and 9 units. Find its area.
s = (7 + 8 + 9) / 2 = 24 / 2 = 12
A = √[12(12 − 7)(12 − 8)(12 − 9)]
A = √[12 × 5 × 4 × 3]
A = √720
A ≈ 26.83 square units
Example 2
A triangular plot of land has sides of 150 m, 200 m, and 250 m. Find the area in square meters.
s = (150 + 200 + 250) / 2 = 600 / 2 = 300
A = √[300(300 − 150)(300 − 200)(300 − 250)]
A = √[300 × 150 × 100 × 50]
A = √225,000,000
A = 15,000 m²
Example 3 — Checking for a Valid Triangle
Can a triangle have sides 3, 4, and 5? Verify with Heron's formula.
Triangle inequality: 3 + 4 > 5 ✓, 3 + 5 > 4 ✓, 4 + 5 > 3 ✓
s = (3 + 4 + 5) / 2 = 6
A = √[6(6 − 3)(6 − 4)(6 − 5)] = √[6 × 3 × 2 × 1] = √36
A = 6 square units (this is a right triangle: 3-4-5)
When to Use It
- When you know all three sides but not the height
- Surveying and land measurement (irregular plots)
- Calculating areas of irregular shapes by dividing them into triangles
- 3D modeling and computer graphics (mesh surface area)