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
Key Notes
- Formula: Area = √(s(s−a)(s−b)(s−c)): Where s = (a+b+c)/2 is the semi-perimeter. Only the three side lengths are needed — no angles, no height measurement required. This makes it uniquely useful when all sides are measured but angles are unavailable.
- Numerically stable version for thin triangles: For a very flat triangle (nearly degenerate), Heron's formula can suffer catastrophic cancellation in floating-point arithmetic. The numerically robust form: Area = ¼√((a+b+c)(−a+b+c)(a−b+c)(a+b−c)) or use the sorted-side formula to avoid precision loss.
- Derived from the Law of Cosines: Heron's formula can be derived by expressing cosC via the Law of Cosines, then substituting into Area = ½ab sinC and simplifying. The result is a purely algebraic expression in the three side lengths.
- Triangle inequality check: For a valid triangle, each side must be less than the sum of the other two. If not, s−a, s−b, or s−c will be negative, making the term under the square root negative — a signal of an invalid triangle, not a real area.
- Applications: Heron's formula is used in land surveying (computing plot areas from distance measurements), computational geometry (area of triangular mesh faces from vertex coordinates), GIS (geographic area calculations), and ancient civil engineering before trigonometry was systematized.