Heron's Formula Calculator
Calculate triangle area from three side lengths using Heron's formula, with validation plus perimeter and semi-perimeter in your chosen unit.
Heron's Formula Calculator
Calculate triangle area from three side lengths using Heron's formula, with validation plus perimeter and semi-perimeter in your chosen unit.
About Heron's formula
Heron's formula is one of the most elegant results in elementary geometry. It lets you compute the area of a triangle using only the three side lengths, without needing a height or an angle. If the sides are a, b, and c, first compute the semi-perimeter s = (a + b + c) / 2. Then the area is √(s(s-a)(s-b)(s-c)). This calculator applies that formula directly and also reports the perimeter and semi-perimeter so you can inspect every intermediate quantity.
The formula is especially helpful in real situations where you can measure edges more easily than altitudes. Surveying, construction layout, fabrication, robotics, and computer graphics often produce side lengths first. When the three sides are known, Heron's formula gives the area in one step. That makes it valuable both for hand calculations and for automated geometry workflows.
Before using the formula, the side lengths must satisfy the triangle inequality: the sum of any two sides must be greater than the third side. If that condition fails, the three segments cannot close into a triangle at all. The calculator checks this explicitly, because otherwise the expression under the square root becomes zero or negative and no valid triangle area exists. This validation is not just a programming detail; it reflects the geometry itself.
Units matter too. The side lengths can be in meters, centimeters, millimeters, feet, inches, or yards, and the calculator keeps the area and linear measures consistent with that choice. Perimeter and semi-perimeter stay in the original unit, while area is reported in squared units. If the sides are entered in centimeters, the area is in square centimeters. Mixing units would invalidate the result, so the safest practice is to convert all sides first and then calculate.
Heron's formula also provides insight into special triangles. A 3-4-5 triangle gives an area of 6, while a 13-14-15 triangle gives 84. Equilateral triangles, isosceles triangles, and many scalene triangles all fit the same formula, which is part of its appeal. Use this calculator when you know the three side lengths and want a fast, reliable area calculation without introducing extra trigonometry or coordinate geometry.
Heron's formula examples
These examples show how the same formula handles familiar right triangles and general scalene triangles.
| Input | Output | Notes |
|---|---|---|
| a = 3, b = 4, c = 5 | Area = 6, perimeter = 12, s = 6 | This classic right triangle produces an exact integer area. It is a good quick check for any Heron's formula implementation. |
| a = 13, b = 14, c = 15 | Area = 84, perimeter = 42, s = 21 | A famous scalene triangle with an exact integer area. The semi-perimeter makes the square-root expression especially neat. |
| a = 7.5, b = 8.2, c = 9.1 | Area ≈ 29.019538, perimeter = 24.8, s = 12.4 | Decimal side lengths work naturally. This is useful for measured geometry rather than textbook integer examples. |
How to use Heron's formula calculator
- Enter the three side lengths of the triangle.
- Choose the unit that matches all three side measurements.
- Click "Calculate Area" to compute the area, perimeter, and semi-perimeter.
- Use "Reset" to clear the inputs and start a new triangle.
Heron's formula FAQ
When should I use Heron's formula?
Use Heron's formula whenever you know all three side lengths but do not know a height or included angle. It is one of the most direct ways to find the area of a triangle from side-only measurements.
What is the semi-perimeter?
The semi-perimeter is half the perimeter: s = (a + b + c) / 2. It appears naturally in Heron's formula and is often useful on its own in geometry problems.
Why does the calculator check the triangle inequality?
Three segments form a triangle only if every pair sums to more than the remaining side. If that rule fails, no geometric triangle exists, so an area calculation would be meaningless.
What unit is the area shown in?
The area is shown in squared units based on the side-length unit you selected. For example, if the sides are in feet, the area is reported in square feet, while perimeter and semi-perimeter remain in feet.