Heron's Formula Calculator

Heron's formula gives the area of a triangle from its three side lengths a, b, and c without needing any angle or height. First compute the semiperimeter s = (a + b + c) / 2, then the area is the square root of s times (s - a) times (s - b) times (s - c). It is named after Heron of Alexandria, who recorded it around 60 AD.

The semiperimeter s is half the triangle's perimeter, so s = (a + b + c) / 2. Heron's formula is written in terms of s because the area depends symmetrically on how much each side falls short of the half-perimeter. The three terms (s - a), (s - b), and (s - c) are each positive exactly when the sides can form a real triangle.

Three lengths form a triangle only if they satisfy the triangle inequality: each side must be strictly shorter than the sum of the other two (a + b > c, b + c > a, and a + c > b). If the longest side equals or exceeds the sum of the other two, the sides cannot close into a triangle. This calculator checks that first and tells you when no valid triangle exists rather than returning a meaningless area.

The area is unitless in the sense that it inherits whatever unit you enter the sides in. If you type the sides in centimeters, the area is in square centimeters; if you type inches, it is in square inches. Just make sure all three sides use the same unit, because mixing units gives a wrong result.

The inradius r is the radius of the largest circle that fits inside the triangle, computed as the area divided by the semiperimeter (r = area / s). The circumradius R is the radius of the circle passing through all three vertices, computed as the product of the sides divided by four times the area (R = abc / 4A). Both follow directly once Heron's formula gives you the area.

Yes. Heron's formula works for every triangle, including right, acute, and obtuse ones. For a 3-4-5 right triangle the semiperimeter is 6 and the area is the square root of 6 times 3 times 2 times 1, which equals 6, the same answer you get from one half base times height (one half times 3 times 4).