Heron's Formula Calculator: Find Triangle Area from Three Sides (2026)
Calculate any triangle's area using Heron's formula. Enter three side lengths to find area, semi-perimeter, and perimeter with step-by-step solutions.
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Put these formulas into practice with our instant, step-by-step Heron's Formula Calculator.
Heron\'s formula is one of the most elegant results in classical geometry, allowing you to calculate the area of any triangle using only its three side lengths. Named after Heron of Alexandria, a Greek mathematician who lived around 10-70 AD, this formula eliminates the need to know the height or angles of the triangle. Our free Heron\'s formula calculator makes these computations instant and accurate.
Key Takeaway
Heron\'s formula calculates the area of any triangle from its three sides: A = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 is the semi-perimeter.
What is Heron\'s Formula?
Heron\'s formula states that the area of a triangle with sides a, b, and c is:
where s = (a + b + c) / 2 (the semi-perimeter)
The formula works for any triangle — scalene, isosceles, or equilateral. It is particularly useful when you know all three side lengths but not the height, which is often the case in practical applications like surveying and land measurement.
How Heron\'s Formula Works
The formula involves three steps:
- Calculate the semi-perimeter: s = (a + b + c) / 2
- Compute the product: s(s-a)(s-b)(s-c)
- Take the square root: A = √[s(s-a)(s-b)(s-c)]
Worked Example: Triangle with Sides 5, 12, 13
Let\'s compute the area of a triangle with sides a = 5, b = 12, c = 13:
| Step | Calculation | Result |
|---|---|---|
| Semi-perimeter | (5 + 12 + 13) / 2 | 15 |
| s - a | 15 - 5 | 10 |
| s - b | 15 - 12 | 3 |
| s - c | 15 - 13 | 2 |
| Product | 15 × 10 × 3 × 2 | 900 |
| Area | √900 | 30 |
The area is 30 square units. Note that this is a right triangle (5² + 12² = 25 + 144 = 169 = 13²), so we could also compute the area as (1/2) × 5 × 12 = 30, confirming our result.
The Triangle Inequality Theorem
Before applying Heron\'s formula, verify that the three sides form a valid triangle. The triangle inequality theorem states that the sum of any two sides must be greater than the third side:
- a + b > c
- a + c > b
- b + c > a
If any of these conditions fail, the three lengths cannot form a triangle, and Heron\'s formula will give a negative number under the square root (or zero for a degenerate triangle).
Real-World Applications
Heron\'s formula is widely used in:
- Surveying: Calculating land areas from boundary measurements
- Architecture: Computing triangular floor plans and roof sections
- Engineering: Analyzing triangular structural elements
- Navigation: Triangulation for GPS and mapping
- Computer Graphics: Mesh generation and area calculations
Frequently Asked Questions
What is the semi-perimeter?
The semi-perimeter is half the perimeter of the triangle: s = (a+b+c)/2. It simplifies the area formula and is used in many geometric calculations. For sides 3, 4, 5, the semi-perimeter is (3+4+5)/2 = 6.
When should I use Heron\'s formula?
Use Heron\'s formula when you know all three side lengths but not the height. It works for any triangle — scalene, isosceles, or equilateral. It is particularly useful when the height is difficult to measure directly.
Who was Heron of Alexandria?
Heron of Alexandria was a Greek mathematician and engineer who lived around 10-70 AD. He made contributions to geometry, mechanics, and pneumatics. The formula for triangle area is attributed to his work 'Metrica'.
Try our Heron\'s formula calculator to compute triangle areas instantly, or explore our triangle perimeter calculator and equilateral triangle calculator for more triangle computations.