Heron's Formula Calculator

Calculate the area of any triangle from its three side lengths using Heron's formula with step-by-step solutions.

Enter Triangle Side Lengths

Result

Triangle Area (Heron's Formula)
14.696938
square units
Side a 5
Side b 6
Side c 7
Semi-perimeter (s) 9
Perimeter 18
Triangle Type Scalene
Height (to side a) 5.878775

Step-by-Step Solution

A = sqrt(s(s-a)(s-b)(s-c))

What Is Heron's Formula?

Heron's formula (also known as Hero's formula) is a mathematical formula that allows you to calculate the area of a triangle when you know the lengths of all three sides. Named after Hero of Alexandria, a Greek mathematician and engineer from the 1st century AD, this formula is particularly useful when the height of the triangle is unknown.

The Formula

Heron's Formula

Where s is the semi-perimeter: s = (a + b + c) / 2

A = sqrt(s(s-a)(s-b)(s-c))

Alternative Form

Expanded without the semi-perimeter variable.

A = (1/4)sqrt((a+b+c)(-a+b+c)(a-b+c)(a+b-c))

Numerical Stability Form

When a >= b >= c, this form avoids floating-point errors.

A = (1/4)sqrt((a+(b+c))(c-(a-b))(c+(a-b))(a+(b-c)))

Triangle Inequality Theorem

For three lengths to form a valid triangle, the triangle inequality must hold: the sum of any two sides must be greater than the third side. This calculator automatically validates this condition before computing the area.

  • a + b > c
  • a + c > b
  • b + c > a

Triangle Classification by Sides

  • Equilateral: All three sides are equal (a = b = c)
  • Isosceles: Exactly two sides are equal
  • Scalene: All three sides are different

Step-by-Step Example

Find the area of a triangle with sides a = 5, b = 6, c = 7:

  1. Calculate the semi-perimeter: s = (5 + 6 + 7) / 2 = 9
  2. Compute each factor: s - a = 4, s - b = 3, s - c = 2
  3. Multiply under the radical: 9 x 4 x 3 x 2 = 216
  4. Take the square root: A = sqrt(216) = 14.6969 square units

History of Heron's Formula

Hero of Alexandria first described this formula in his work Metrica (circa 60 AD). However, some historians believe that Archimedes may have known the formula two centuries earlier. The formula has been independently discovered by mathematicians in China and India as well, demonstrating its fundamental nature in geometry.

When to Use Heron's Formula

  • When you know all three sides of a triangle but not the height
  • When the triangle is irregular and standard base-height calculations are impractical
  • In surveying and navigation where distances between points are measured directly
  • In computer graphics for calculating areas of triangulated surfaces