1st-century math flex
Heron wrote this formula around 60 CE, but historians think Archimedes knew it earlier—meaning the shortcut is over two millennia old and still the go-to for SSS area.
Ancient yet modern
Heron’s Formula Calculator — Triangle Area from 3 Sides
Diagram & Inputs
Tip: All three sides must satisfy the triangle inequality (each side < sum of the other two).
Results
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.
5 Fun Facts about Heron’s Formula
Perimeter → max area
Fix the perimeter and Heron shows the equilateral triangle wins for area. With total perimeter P, the max area is \(P^2\sqrt{3}/36\)—a tidy square-law result.
Area champion
3-4-5 hides a pattern
The classic 3-4-5 right triangle has semiperimeter 6, area 6, and inradius 1. One glance at Heron’s square-root factors shows why those numbers line up so cleanly.
Pythagorean delight
Skinny triangles fool floats
When a triangle is nearly flat, s-a, s-b, and s-c get tiny and subtraction can lose precision. A stable variant (Kahan’s rearrangement) keeps the digits you’d otherwise throw away.
Numerical sanity
It scales to quadrilaterals
Brahmagupta’s formula is Heron’s with four sides—but it only works when the quadrilateral is cyclic. Bretschneider’s formula adds the angles to handle any four-sider.
Beyond triangles
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.