Financial term · Reviewed August 17, 2026
Heron's formula
Heron's formula: A method for calculating a triangle's area from its three side lengths alone, without needing a separate height measurement. The method computes the semi-perimeter, half the sum of the three sides, then combines it with each side under a square root. Unlike the base-times-height formula, it applies to any triangle. Right, obtuse or scalene. The only condition is that the three lengths can actually form one. The Triangle Area (Heron's Formula) Calculator returns this figure from heron's formula: s = (a+b+c)/2; area = √(s(s−a)(s−b)(s−c)). At that calculator’s documented worked-example inputs the area is 14.7; holding every other input steady, moving side a from 4 to 6 moves it from 12 to 17.1, a spread of 5.1.
Three sides, no height, and you still get the area. That solves the common practical case exactly: an irregular plot of land where distances can be paced or measured but no perpendicular height exists to measure from. Base times height is useless there, and Heron's formula is not.
| Term | Heron's formula |
|---|---|
| Category | Geometry |
| How it is found | Heron's formula: s = (a+b+c)/2; area = √(s(s−a)(s−b)(s−c)) |
| Worked example | Area: 14.7 |
| Computed by | Triangle Area (Heron's Formula) Calculator |
| Related terms | 2 defined in this glossary |
Before you rely on it
What to check
Check the three lengths can actually form a triangle.
The common error
Where people go wrong with heron's formula
Feeding in measurements where one side exceeds the other two combined. No such triangle exists, and the formula fails at the square root , not warning you the input was impossible.
Worked through
Heron's formula in plain numbers
A triangle has sides of 13, 14 and 15 units. The semi-perimeter s is (13 + 14 + 15) ÷ 2, or 21; the area is the square root of s × (s − a) × (s − b) × (s − c). That is the square root of 21 × 8 × 7 × 6, which is the square root of 7,056, or 84 square units. No angle and no height were needed, only the three sides, and check it with a base of 14 and a height of 12, where ½ × 14 × 12 also gives 84. The formula is handy when a plot of land is measured only along its edges.
Easily confused
Heron's formula and its neighbours
Heron's formula is not the same as half base times height, though both give the area; the base-height version needs a perpendicular height, which is often hard to measure. Heron's formula uses only side lengths. It also differs from the law of cosines and the law of sines, which relate sides and angles, and from Brahmagupta's formula for cyclic quadrilaterals, which extends the idea. The three sides must satisfy the triangle inequality, so that any two add up to more than the third, or the square root has no real value.
In real decisions
Where heron's formula turns up
Surveyors use it to find the area of a triangular lot from measured sides. Builders and landscapers use it to estimate materials for triangular sections; students use it in geometry. Computer graphics and mapping software use versions of it to compute areas of triangles on a mesh; it turns a physical measurement task, walking the three edges, into a straightforward calculation without any need to establish a right angle.
Know the edges
Where heron's formula stops being reliable
Numerical accuracy suffers for very thin triangles, where the terms nearly cancel, and stable versions of the formula are used in software. Measurement errors in the sides carry through to the area. It applies to flat triangles on a plane. On curved surfaces, such as large areas of the earth, spherical formulas are needed. For real land measurements, official surveys should be used for legal purposes.
Quick answers
What people ask about heron's formula
Does it work for any triangle? Yes, as long as the three sides can form a triangle. Any two sides must add up to more than the third.
What if the triangle is very thin? Rounding errors can grow because two terms nearly cancel. Software uses a rearranged version that is more stable.
Put it to work
How heron's formula is calculated
At the triangle area (heron's formula) calculator’s worked-example inputs, area is 14.7, perimeter is 18. Moving side a from 4 to 6 moves that first figure from 12 to 17.1.
Open the Triangle Area (Heron's Formula) Calculator →
Formulas assume ideal geometric shapes. Real-world measurements carry their own error, and every input must use the same unit: mixing feet and inches silently produces a wrong answer.
Keep reading
Nearby in the glossary
- ← Heat index
- Home equity →
- All 174 defined termsThe complete Free Calculators Online financial glossary.