Heron’s formula gives the area of a triangle from its sides alone, with no angles or heights needed. Surveyors use it to find the area of a triangular plot from three tape measurements; it dates from Heron of Alexandria around 60 AD.
How to use the Heron’s Formula Calculator
- Side a
- Side b
- Side c
Results update as you type, and everything is calculated in your browser.
How it works
Enter the three sides. The semi-perimeter is half the perimeter; if the sides break the triangle inequality, no triangle exists and the area is undefined.
Inputs explained
| Field | What to enter |
|---|---|
| Side a | Units: cm, m, mm, in, ft, yd … |
| Side b | Units: cm, m, mm, in, ft, yd … |
| Side c | Units: cm, m, mm, in, ft, yd … |
What each result means
- Semi-perimeter s = ([Side a] + [Side b] + [Side c])÷2
- Area = sqrt([Semi-perimeter s]×([Semi-perimeter s] – [Side a])×([Semi-perimeter s] – [Side b])×([Semi-perimeter s] – [Side c]))
- Perimeter = [Side a] + [Side b] + [Side c]
- Height to side a = 2×[Area]÷[Side a]
- Height to side b = 2×[Area]÷[Side b]
- Height to side c = 2×[Area]÷[Side c]
- Inradius = [Area]÷[Semi-perimeter s]
- Circumradius = [Side a]×[Side b]×[Side c]÷(4×[Area])
Example values
How the main result changes when Side a changes and everything else stays at the example values above:
| Side a (cm) | Area (cm²) |
|---|---|
| 2.5 | 7.3077 |
| 3.75 | 11.2469 |
| 5 | 14.6969 |
| 6.25 | 17.5815 |
| 7.5 | 19.7317 |
| 10 | 20.6625 |
Formula
A = √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2
Worked example
Inputs: Side a: 5 cm; Side b: 6 cm; Side c: 7 cm.
Results: Semi-perimeter s = 9 cm; Area = 14.696938 cm²; Perimeter = 18 cm; Height to side a = 5.878775 cm; Height to side b = 4.898979 cm; Height to side c = 4.199125 cm; Inradius = 1.632993 cm; Circumradius = 3.572173 cm.
Key terms
- Semi-perimeter
- Half the perimeter.
Frequently asked questions
Does Heron's formula work for every triangle?
Yes, as long as the sides form a triangle.
What is the semi-perimeter?
Half of the perimeter.
What if the result is undefined?
The sides fail the triangle inequality.
Is there a version for quadrilaterals?
Brahmagupta's formula for cyclic quadrilaterals.