Heron's Formula Calculator

About Heron's Formula Calculator

The Heron's Formula Calculator computes the area of any triangle when you know all three side lengths. Enter sides \(a\), \(b\), and \(c\), and instantly get the area using Heron's formula \(A = \sqrt{s(s-a)(s-b)(s-c)}\) where \(s = \frac{a+b+c}{2}\) is the semi-perimeter. The calculator also provides the perimeter, all three altitudes, interior angles, inradius, circumradius, and triangle classification with step-by-step formulas and an interactive diagram.

What Is Heron's Formula?

Heron's formula (sometimes called Hero's formula) is named after Hero of Alexandria, a Greek mathematician who lived in the 1st century AD. It allows you to calculate the area of a triangle using only the three side lengths — no angles or heights needed. The formula is:

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

where \(s = \frac{a+b+c}{2}\) is the semi-perimeter (half the perimeter). This elegant formula works for all types of triangles — equilateral, isosceles, scalene, acute, right, and obtuse.

Real-World Applications

Key Formulas

How to Use the Heron's Formula Calculator

1. Enter side lengths: Type the three side lengths (a, b, c) of your triangle. You can use decimal values or click a quick example button to auto-fill sample values.
2. Preview the triangle: As you type, the live triangle preview updates in real-time, showing the actual shape and proportions along with a quick area estimate.
3. Click Calculate Area: Press the button to compute all results. The calculator validates the triangle inequality automatically.
4. Review results: See the area, perimeter, semi-perimeter, all three altitudes, interior angles, inradius, circumradius, and triangle classification. Use the diagram toggles to show or hide heights, angles, the incircle, and circumcircle.

Triangle Inequality Theorem

Not every combination of three positive numbers can form a triangle. The triangle inequality theorem requires that the sum of any two sides must be greater than the third side: \(a + b > c\), \(a + c > b\), and \(b + c > a\). If any of these conditions is not met, the three lengths cannot form a valid triangle. This calculator automatically checks this condition and displays an error message if the sides are invalid.

Triangle Types

Triangles can be classified by their sides and angles. By sides: an equilateral triangle has all three sides equal, an isosceles triangle has exactly two sides equal, and a scalene triangle has all three sides different. By angles: an acute triangle has all angles less than 90°, a right triangle has one angle exactly 90°, and an obtuse triangle has one angle greater than 90°. This calculator determines both classifications automatically.

Inradius and Circumradius

The inradius (\(r\)) is the radius of the inscribed circle — the largest circle that fits inside the triangle, tangent to all three sides. It is calculated as \(r = A/s\). The circumradius (\(R\)) is the radius of the circumscribed circle — the circle passing through all three vertices. It is calculated as \(R = abc/(4A)\). These two radii are related by Euler's formula: the distance between the incenter and circumcenter is \(\sqrt{R^2 - 2Rr}\).

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