Heron's Formula Calculator — Area of a Triangle with 3 Sides
If you want to calculate the area of a triangle when you know the lengths of all three sides, you're in the right place. This Heron's formula calculator does exactly that — simply enter sides a, b, and c, choose your units, and get the area instantly. You also get the perimeter, semiperimeter, all three altitudes, all three angles, the circumradius, and the inradius.
Heron's Area Formula
Heron's formula (also called Hero's formula) gives the area of a triangle from its three side lengths alone — no angles needed. It was described by Hero of Alexandria in his book Metrica (ca. 60 AD):
A = √(s(s − a)(s − b)(s − c))
where s is the semiperimeter:
s = (a + b + c) / 2
Alternative forms
If you prefer a version without the semiperimeter intermediate step:
A = (1/4) × √((a+b+c)(−a+b+c)(a−b+c)(a+b−c))
or equivalently:
A = (1/4) × √(4a²b² − (a² + b² − c²)²)
Heron's Formula Proof
The proof starts from the standard area formula A = ½ b h, where b is the base and h is the corresponding height. Using the Pythagorean theorem on the two right sub-triangles formed by the altitude, one can express h entirely in terms of a, b, and c. Substituting and simplifying leads directly to Heron's formula.
How to Use This Heron's Formula Calculator
- Select your measurement system — Metric (mm, cm, m, km) or US Customary (in, ft, yd, mi).
- Choose the unit that matches your measurements.
- Enter the three side lengths — sides a, b, and c of your triangle.
- Click Calculate Area.
The calculator verifies the triangle inequality (the sum of any two sides must exceed the third) and returns:
- Area
- Perimeter and semiperimeter
- All three altitudes (heights)
- All three interior angles
- Circumscribed circle radius (R)
- Inscribed circle radius (r)
Frequently Asked Questions
What is Heron's formula?
Heron's formula calculates the area of any triangle when the three side lengths are known: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2.
Can I use Heron's formula for any triangle?
Yes — it works for scalene, isosceles, equilateral, and right triangles alike. The only requirement is that the three sides must satisfy the triangle inequality.
What is the semiperimeter?
The semiperimeter is half the perimeter: s = (a + b + c) / 2. It is the key intermediate value in Heron's formula.
What units does this calculator support?
Metric: millimeters (mm), centimeters (cm), meters (m), kilometers (km).
US Customary: inches (in), feet (ft), yards (yd), miles (mi).
What if my sides don't form a valid triangle?
If the triangle inequality is violated (e.g., a + b ≤ c), the calculator will display an error message. A valid triangle requires: a + b > c, a + c > b, and b + c > a.