What is Heron’s formula?

A way to get a triangle’s area from its three side lengths alone, with no angle and no perpendicular height. Take half the perimeter, subtract each side from it in turn, multiply the four numbers together and take the square root. It is roughly two thousand years old and still the practical method for an irregular plot of land.

Why do my two answers disagree?

Because at least one measurement is wrong, and that is useful information rather than a bug. Most often the "height" was measured along a side instead of perpendicular to the base. The implied height printed here is what your three sides say the perpendicular should be — compare it with what you measured.

What if the three sides cannot make a triangle?

Then Heron’s formula returns zero here rather than an error. Any two sides must add up to more than the third; 3, 4 and 20 describe nothing that closes. If you get zero, re-measure — one of the three is wrong.

Math

Triangle Area Calculator

Two ways to the same answer, depending on which measurements you actually have.

The formula

area = base × height ÷ 2

s = (a + b + c) ÷ 2

area = √(s × (s − a) × (s − b) × (s − c)) — Heron

height on a = 2 × area ÷ a

Worked example

A 12-5-13 right triangle

base × height: 12 × 5 ÷ 2 = 30
s = (12 + 13 + 5) ÷ 2 = 15
Heron: √(15 × 3 × 2 × 10) = √900 = 30
Both routes agree, which is the check

Where this goes wrong

Measuring the height along a sloping side

The height in base × height ÷ 2 is the perpendicular distance from the base to the opposite corner, not the length of a side. On a triangle with a 5 m base and sides of 4 m leaning at 30°, using the side gives 10 m² where the true area is 5 m² — double. If you can only measure the three sides, use Heron’s formula instead, which needs no perpendicular at all. This page runs both and prints the height the sides imply, so a mismatch is visible.

Questions

What is Heron’s formula?: A way to get a triangle’s area from its three side lengths alone, with no angle and no perpendicular height. Take half the perimeter, subtract each side from it in turn, multiply the four numbers together and take the square root. It is roughly two thousand years old and still the practical method for an irregular plot of land.
Why do my two answers disagree?: Because at least one measurement is wrong, and that is useful information rather than a bug. Most often the "height" was measured along a side instead of perpendicular to the base. The implied height printed here is what your three sides say the perpendicular should be — compare it with what you measured.
What if the three sides cannot make a triangle?: Then Heron’s formula returns zero here rather than an error. Any two sides must add up to more than the third; 3, 4 and 20 describe nothing that closes. If you get zero, re-measure — one of the three is wrong.