Triangle Area Explained: Base × Height and Heron’s Formula

Ask for a triangle's area and you get a follow-up question: what do you know about it? A base with a perpendicular height leads to one formula; three bare side lengths lead to another, older and cleverer. A triangle area calculator needs both, because real-world triangles rarely come with their height labeled.

Route one: base times height, halved

When a base and its perpendicular height are known, the area is half the containing rectangle:

area = (base × height) / 2

base 10, height 6  →  (10 × 6) / 2 = 30
base 7.5, height 4.2  →  15.75

Any side can serve as the base — the height is simply the perpendicular distance from that side to the opposite vertex. Choose a different base and the height changes to compensate, so the area never moves. The catch is practical, not mathematical: the height must be perpendicular to the base, and on an arbitrary scalene triangle there may be no way to measure it. That is where route two comes in.

Route two: Heron's formula from three sides

Known in Alexandria around 2,000 years ago, Heron's formula builds the area from the side lengths alone. First collapse the perimeter into the semi-perimeter s, then multiply and take the square root:

s = (a + b + c) / 2
area = √( s(s−a)(s−b)(s−c) )

The 3-4-5 right triangle shows the machinery at its cleanest: s = (3+4+5)/2 = 6, so the product is 6 × 3 × 2 × 1 = 36 and the area is √36 = 6 — an exact integer hiding inside a square-root formula.

Heron's formula on less tidy triangles

Change the sides to 5, 6, 7 and the rounding disappears: s = 9, the product is 9 × 4 × 3 × 2 = 216, and the area is √216 = 14.6969384567. An equilateral triangle with all sides 6 gives s = 9 and 9 × 3 × 3 × 3 = 243, so the area is √243 = 15.5884572681 — which you can cross-check against the equilateral shortcut (√3/4)·a² ≈ 0.4330 × 36 ≈ 15.59. Two independent formulas agreeing is the best sanity check in mathematics, and the calculator runs Heron's version of it on every three-side input.

The triangle inequality guard

Not every triple of positive numbers forms a triangle: the two shorter sides must beat the longest one, a + b > c (and the same for the other pairings). Sides 1, 2, and 10 fail because 1 + 2 < 10 — no triangle can close. The check matters computationally as well as physically: for a near-degenerate triple, one of (s−a), (s−b), (s−c) goes negative and Heron's product goes negative with it, demanding the square root of a negative number. The Triangle Area Calculator validates the inequality before computing, so a bad triple produces an error message instead of NaN.

Units, and how this differs from a full triangle solve

Enter every length in one unit and the area comes back in that unit squared: centimeters in, square centimeters out. Mixing units silently scales one side relative to the others and corrupts the answer, so convert first. And note the boundary of this tool's job: it answers exactly one question — the area — by two routes. The Triangle Calculator is the broader instrument: given the three sides it also returns the perimeter, all three angles, and the triangle's classification. If you need the area plus the angles, use the full solver; if you need the area fast, from base-and-height or sides alone, the area calculator is the shorter path.

Try it yourself

The Triangle Area Calculator handles both routes — base × height ÷ 2 and Heron's formula — with the semi-perimeter shown for the three-sides method. The Triangle Calculator solves everything about a 3-side triangle, and for the right-triangle special case the reasoning traces back to how the Pythagorean theorem works.