Heron’s Formula Calculator — Triangle Area from 3 Sides

Heron’s Formula Explained

Heron’s formula computes the area of a triangle using only its side lengths. If the sides are \(a\), \(b\), and \(c\), define the semiperimeter \(s = \tfrac{a+b+c}{2}\). The area is then \(A = \sqrt{s(s-a)(s-b)(s-c)}\). This is especially convenient when you don’t have the triangle’s height or angles.

The calculator follows a clear process: (1) it reads your three side lengths, (2) checks the triangle inequality to ensure a valid triangle exists, (3) computes \(s\) and the area \(A\), and (4) derives useful secondary values. You’ll see the perimeter \(P = a+b+c\), the inradius \(r = A/s\) (radius of the inscribed circle), the circumradius \(R = \tfrac{abc}{4A}\) (radius of the circumscribed circle), and the three altitudes \(h_a = \tfrac{2A}{a}\), \(h_b = \tfrac{2A}{b}\), \(h_c = \tfrac{2A}{c}\). Lengths are labeled in your chosen unit, while area uses the squared unit automatically.

Why does Heron’s formula work? One geometric route uses the identity \(A = \tfrac{1}{2}ab\sin C\) and the Law of Cosines \(\cos C = \tfrac{a^2+b^2-c^2}{2ab}\) to eliminate the angle \(C\), leading after some algebra to the symmetric expression under the square root. In practice, Heron is robust and fast for real-world inputs such as survey measurements, CAD sketches, and classroom exercises. For extremely skinny triangles or very large/small numbers, rounding can accumulate; you can adjust the decimal-places setting to suit your needs.

Everything runs entirely in your browser (client-side JavaScript), so inputs never leave your device.

Heron’s Formula: FAQs

What inputs are required?

Three side lengths \(a, b, c\) that satisfy the triangle inequality. Negative or zero values are invalid.

What else does the tool compute?

Semiperimeter, perimeter, inradius, circumradius, and the three altitudes.

Is my data private?

Yes—100% client-side.