Calculate the area of any triangle from its three side lengths using Heron's formula with semi-perimeter breakdown.
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Result
Area
Perimeter
12
Semi-perimeter (s)
6
Also in Triangles
Geometry
The Heron's formula calculator finds the area of any triangle when you know all three side lengths. It computes the semi-perimeter, verifies the triangle inequality, and returns the exact area without needing height or angle measurements.
Heron's formula calculates triangle area from the three sides a, b, and c. First compute the semi-perimeter s = (a + b + c) / 2, then the area is sqrt(s * (s-a) * (s-b) * (s-c)).
The formula is attributed to Hero of Alexandria (first century AD) and works for any valid triangle, including obtuse and scalene triangles where the height is not obvious.
s = (a + b + c) / 2
Semi-perimeter.
A = sqrt(s(s-a)(s-b)(s-c))
Area from the semi-perimeter and three sides.
Three lengths form a valid triangle only if each side is less than the sum of the other two: a < b + c, b < a + c, and c < a + b. If any condition fails, no triangle exists and the formula produces an imaginary result.
The calculator checks this condition before computing and reports an error if the sides cannot form a triangle.
Frequently asked questions
Use Heron's formula when you know all three side lengths but not the height. It avoids the need to compute an altitude first.
Yes. For a right triangle with legs a, b and hypotenuse c, Heron's formula gives the same result as (1/2) * a * b, though the simpler formula is faster.
Half the perimeter: s = (a + b + c) / 2. It simplifies the area formula and appears in many other triangle identities.
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Find the height of a triangle from its area and base, or from side lengths using Heron's formula.
Calculate the perimeter of a triangle from its three side lengths with semi-perimeter and individual side breakdown.
Calculate the area, perimeter, height, inradius, and circumradius of an equilateral triangle from its side length.
Solve for the missing side of a right triangle using the Pythagorean theorem, with perimeter and area included.
