Heron's Formula Calculator
Calculate the area of a triangle from its three side lengths alone — no angle or height needed.
How to use Heron's formula calculator
- Enter the three side lengths of the triangle.
- The calculator works out the area automatically — no angle or height is needed.
- The sides must be able to form a valid triangle, otherwise "Invalid" is shown.
Heron's formula
Heron's formula gives the area of any triangle directly from its three side lengths a, b and c. First the semi-perimeter (s) is calculated, then the area itself:
Area = √(s × (s − a) × (s − b) × (s − c))
Worked example: a triangle with sides 3, 4 and 5 has a semi-perimeter of s = (3 + 4 + 5) ÷ 2 = 6, and the area comes to √(6 × (6−3) × (6−4) × (6−5)) = √36 = 6. The appeal of Heron's formula is that it needs no height: for this right triangle you could also compute 3 × 4 ÷ 2 = 6, but Heron works equally well on triangles where no height is known.
The formula is named after Heron of Alexandria, who described it around 60 AD — though there are indications it was known earlier. It's especially useful when you know all three sides but not any angle or height, since the alternative (½ × base × height) isn't directly applicable then.
Frequently asked questions
When should I use Heron's formula instead of ½ × base × height?
Use Heron's formula when you know all three side lengths but not the height. If you know the base and height directly instead, ½ × base × height is both simpler and faster.
Why does the calculator show "Invalid"?
Because the three sides can't form a valid triangle — the sum of the two shortest sides must always be greater than the longest side (the triangle inequality). If it's equal to or less, the triangle "flattens" into a straight line or can't close at all.
Does the formula work for all types of triangles?
Yes, Heron's formula works for any triangle — right, isosceles, equilateral, or scalene — as long as the three side lengths are valid.