Library / Foundations: Maths / FMA-35
Foundations: Maths
Heron’s formula
Area of a triangle from its three side lengths alone, with no angle needed.
- Ref
- FMA-35
- Rev
- v1
- Field
- Foundations: Maths
- Result
- A [m²]
a, b and c are the three side lengths in the same unit. s is the semi-perimeter, half the total distance round the triangle. The triangle exists only if every side is shorter than the sum of the other two, which is the same as s − a, s − b and s − c all being positive.
Area of a triangle from its three side lengths alone, with no angle needed. The half-perimeter s must be larger than every side or the triangle cannot exist.
Calculator
- Semi-perimeter s6 m
- Height above side c2.4 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The default 3-4-5 triangle has area 6 m², which agrees with ½ × 3 × 4 for a right-angled triangle, a quick check on the formula.
- The formula needs only side lengths, so it suits a plot measured by tape or pacing where no angles were taken.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Side a | 0.1 to 100 m | 0 | 9.996 | 9.996 | +10% | +9.72 % |
| c Side c | 0.1 to 100 m | 0 | 6 | 6 | +10% | −2.42 % |
| b Side b | 0.1 to 100 m | 1.828 | 7.358 | 5.53 | +10% | +9.11 % |
Sweep of Side b
| Side b (m) | Area (m²) |
|---|---|
| 0.1 | — |
| 0.2371 | — |
| 0.5623 | — |
| 1.334 | — |
| 3.162 | 4.5 |
| 7.499 | 5.035 |
| 17.78 | — |
| 42.17 | — |
| 100 | — |
Sweep one variable
Variables and how each one contributes
Area
What the equation gives you. Everything else feeds it.
Side a
One side of the triangle. Raises the area at the default: a 10 % rise in a (to 3.3 m) raises A by about 10 %. Stretching it towards the sum of the other two sides flattens the triangle and the area falls back to zero.
Side b
A second side of the triangle. Raises the area at the default: a 10 % rise in b raises A by about 9 %. Close to the limit b = a + c the triangle collapses and the area goes to zero.
Side c
The third side of the triangle. Lowers the area at the default, because 5 m is already the longest side: a 10 % rise in c flattens the triangle and lowers A by about 2 %. The area is greatest when the triangle is closest to equilateral.
Worked example with the values above
- Semi-perimeter: s = (3 + 4 + 5)/2 = 6 m.
- Differences: s − a = 3, s − b = 2, s − c = 1. The product s(s − a)(s − b)(s − c) = 36.
- Take the square root: A = 6 m².
Assumptions and limits
- The three sides satisfy the triangle inequality.
- Flat (plane) ground: on a large survey the curvature of the Earth is ignored.
Common mistakes
- Using the perimeter instead of the semi-perimeter s in the formula.
- Mixing units, for example two sides in metres and one in feet, which gives an area in no consistent unit.
Statistics and uncertainty
The area depends on the differences s − a, s − b and s − c, so a nearly flat triangle is very sensitive to length errors: a small error in the longest side can change a thin triangle’s area by tens of per cent. For a well-shaped triangle, a 1 % error in every side gives about 2 % in the area, because area scales with length squared.
Where it is used
- Area of an irregular plot from boundary lengths measured on the ground.
- Cross-checking an area found from a map with coordinates.
Maths you need
- Triangles: perimeter and the triangle inequality
- Square roots: √ of the product