Heron's Formula Calculator
Calculate the area of any triangle from its three side lengths using Heron's formula.
Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2
What Is the Heron's Formula Calculator?
Heron's Formula Calculator computes the area of any triangle given only its three side lengths, with no need for angles or height. Enter sides a, b, and c to get the exact area, heights from each side, angles, and perimeter. The formula works for all valid triangles including obtuse and right triangles.
Formula
How to Use
Enter the three side lengths a, b, and c. The calculator verifies the triangle inequality (each side must be less than the sum of the other two), then computes the semi-perimeter s, and applies Heron's formula. Results include area, all three heights, all three angles, and perimeter.
Example Calculation
Triangle with sides 13, 14, 15: s = (13+14+15)/2 = 21. Area = √(21×8×7×6) = √7056 = 84 sq units. Height from side 14: h = 2×84/14 = 12. This is a classic integer-sided triangle with integer area.
Understanding Heron's Formula
Heron's formula is attributed to Hero of Alexandria (c. 60 AD), though it may have been known to Archimedes centuries earlier. It computes the area of a triangle purely from its three side lengths, making it one of the most practically useful formulas in geometry. The formula: Area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter.
The formula is elegant because it requires only the three side lengths — information that is easy to measure physically (with a tape measure) without needing to establish perpendiculars or measure angles. This makes Heron's formula indispensable in land surveying, architecture, and civil engineering.
Heron's formula extends to non-planar geometry: Bretschneider's formula generalizes it to quadrilaterals, and spherical versions exist for spherical triangles. In computational geometry, it is used in mesh algorithms, finite element analysis, and geographic information systems to compute areas of polygonal regions defined by GPS coordinates.
Frequently Asked Questions
Why is Heron's formula useful?
Heron's formula computes area from side lengths alone, without needing to know any angle or height. This is valuable in surveying, construction, and computational geometry where only distances are measured.
How do I verify three sides form a valid triangle?
The triangle inequality requires each side to be strictly less than the sum of the other two: a < b+c, b < a+c, c < a+b. If any side equals or exceeds the sum of the others, no triangle exists.
Does Heron's formula work for obtuse triangles?
Yes. Heron's formula is valid for all triangles — acute, right, and obtuse. The semi-perimeter s is always greater than each individual side for any valid triangle.
Is there a more numerically stable version?
For near-degenerate triangles (very flat), the formula can lose precision. Kahan's formula rearranges the computation for better numerical stability: √((a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c)))/4.
Is this calculator free?
Yes, completely free with no sign-up required.