How to use the Triangle Area Calculator
- Choose base and height if you can measure a perpendicular drop to one side.
- Choose three sides when you can only measure the edges, which is usual on site.
- Enter the measurements in consistent units.
- Read the area; with three sides you also get the perimeter and semi-perimeter.
- If the result reports an invalid triangle, re-check your measurements against the triangle inequality.
How the calculation works
The base-height formula comes from a simple observation: any triangle is exactly half of a parallelogram with the same base and height. The height must be the perpendicular distance from the base to the opposite vertex, not the length of a slanted side — using a slant length is the single most common error in triangle area work, and it always overstates the result.
Heron's formula solves the practical problem that perpendicular heights are hard to measure in the field, while side lengths are easy. It computes the semi-perimeter, then takes the square root of the semi-perimeter multiplied by its difference from each side. Land surveyors have used it for centuries to break irregular plots into triangles measurable with nothing but a tape.
Heron's formula also validates the input for free. If the three lengths cannot form a triangle — because one side is at least as long as the other two combined — the product under the square root turns negative. That is the triangle inequality expressing itself algebraically, and the calculator reports it as an invalid triangle rather than returning a NaN.
A = ½ × base × height; or with s = (a+b+c)/2, A = √(s(s−a)(s−b)(s−c))Source: Heron's formula, Heron of Alexandria, Metrica (c. 60 CE); base-height formula from Euclid, Elements Book I.
Worked example
A triangular garden bed measures 9 m, 12 m and 15 m along its edges.
- Semi-perimeter: (9 + 12 + 15) / 2 = 18 m.
- Differences: 18 − 9 = 9, 18 − 12 = 6, 18 − 15 = 3.
- Product: 18 × 9 × 6 × 3 = 2,916.
- Area: √2,916 = 54 m².
54 m² — and since 9-12-15 is a scaled 3-4-5, this is a right triangle, confirmed by ½ × 9 × 12 = 54.
Frequently asked questions
Which side counts as the base?+
Any of them, as long as the height is measured perpendicular to that same side. All three choices give the identical area.
Why does my calculation say the triangle is invalid?+
Two of your sides sum to less than or equal to the third, which is geometrically impossible. Re-measure — usually one figure has a transposed digit.
Can I find the area from two sides and the angle between them?+
Yes: half the product of the two sides times the sine of the included angle. The law of cosines calculator reports that figure.
How do I handle an irregular four-sided plot?+
Split it into two triangles along a diagonal, measure that diagonal, and apply Heron's formula to each half.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.