Triangle Area from 3 Sides (Heron's Formula)

Calculate the area of any triangle when you know all three side lengths using Heron's formula.

Enter Side Lengths

a = 5 b = 4 c = 3 h

Result

Enter three side lengths and click Calculate.

Heron's Formula for Triangle Area

The Formula

Heron's formula calculates a triangle's area from its three side lengths a, b, and c. First compute the semi-perimeter:

s = (a + b + c) / 2

Then:

Area = √[s(s - a)(s - b)(s - c)]

Historical Background

Named after Heron of Alexandria (c. 10-70 AD), this formula appeared in his work Metrica. It may have been known to Archimedes centuries earlier. The formula was significant because it computes area from sides alone, without constructing the triangle or measuring its height.

Triangle Inequality

For three lengths to form a valid triangle, they must satisfy the triangle inequality: the sum of any two sides must exceed the third.

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

Special Triangles

  • Equilateral (a = b = c): Area = (√3/4) × a². All angles are 60 degrees.
  • Isosceles (two sides equal): Has a line of symmetry and two equal angles.
  • Right triangle: When a² + b² = c². The classic 3-4-5 triangle is the simplest integer right triangle.
  • Scalene: All sides and angles are different.

Numerical Stability

For very thin triangles, a numerically stable alternative (with a ≥ b ≥ c) is:

Area = ¼ √[(a+(b+c))(c-(a-b))(c+(a-b))(a+(b-c))]

Applications

Heron's formula is widely used in surveying, construction, computer graphics (mesh triangle areas), and any situation where angle measurements are unavailable but side lengths can be measured.