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Geometry

Heron's formula

The area of a triangle from its three side lengths alone, with no height or angle needed.

A=s(s−a)(s−b)(s−c), s=a+b+c2

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

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c = 9b = 8a = 7
  • s = (a + b + c)/2 = 12
  • Area = √(s(s − a)(s − b)(s − c)) = 26.833

What the symbols mean

A
the area of the triangle
a
the first side
b
the second side
c
the third side
s
the semi-perimeter: half of a+b+c

When to use it

Use it when you know all three sides but no height or angle: a triangular plot of land measured with a tape, a roof panel, a triangle given on a map. The sides must satisfy the triangle inequality (the longest side shorter than the other two together) or there is no triangle. The calculator can also work backwards from the area and two sides; there are usually two triangles that fit.

Worked example

A triangular plot has sides of 7 m, 8 m and 9 m. What is its area? (a = 7, b = 8, c = 9)

  1. Heron's formula, where s is the semi-perimeter (half the perimeter):

    A=s(s−a)(s−b)(s−c)

  2. Find the semi-perimeter.

    s=a+b+c2
    =7+8+92
    =15+92
    =242
    =12

  3. Substitute s and the three sides.

    A=12·(12−7)·(12−8)·(12−9)
    =12·5·4·3
    =60·4·3
    =240·3
    =720
    ≈26.8328

Answer s = 12, so A ≈ 26.83 m²

Why it works

Write the area as 12absin⁡C, get cos⁡C from the law of cosines and use sin2⁡C=1−cos2⁡C. After factorising with the difference of two squares, everything collapses into the product s(s−a)(s−b)(s−c). It is named after Heron of Alexandria, who gave a proof in the first century AD.

Common mistakes

  • Using the full perimeter instead of half of it for s.
  • Trying it on sides that cannot form a triangle (such as 1, 2 and 3). One of the brackets becomes 0 or negative.
  • Rounding s before substituting. Keep it exact; it is often a whole number or a half.