Heron's Formula Calculator
Calculate triangle area from three sides with step-by-step solution
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Understanding Heron's Formula
Heron's formula (also known as Hero's formula) is a method for calculating the area of a triangle when you know the lengths of all three sides. It was named after Hero of Alexandria, a Greek mathematician.
The Formula
where s = (a + b + c) / 2
Variables
- a, b, c - The lengths of the three sides of the triangle
- s - The semiperimeter (half the perimeter) of the triangle
- A - The area of the triangle
Triangle Inequality Theorem
For a valid triangle, the sum of any two sides must be greater than the third side:
- a + b > c
- b + c > a
- a + c > b
Frequently Asked Questions
What is Heron's formula used for?
Heron's formula is used to calculate the area of a triangle when you know all three side lengths but don't know the height. It's particularly useful in surveying, navigation, and computer graphics.
Who discovered Heron's formula?
The formula is attributed to Hero (or Heron) of Alexandria, a Greek mathematician and engineer who lived in the 1st century AD. However, it may have been known earlier to Archimedes.
Does Heron's formula work for all triangles?
Yes, Heron's formula works for all types of triangles (scalene, isosceles, and equilateral) as long as the three sides satisfy the triangle inequality theorem.
What is the semiperimeter?
The semiperimeter is half of the triangle's perimeter, calculated as s = (a + b + c) / 2. It's an intermediate value used in Heron's formula.
Can I use Heron's formula for a right triangle?
Yes, but for right triangles, it's usually simpler to use A = (1/2) × base × height. However, Heron's formula will give the same result.
What happens if I enter invalid triangle sides?
The calculator will detect invalid triangles using the triangle inequality theorem and display an error message. Make sure the sum of any two sides is greater than the third side.
How accurate is this calculator?
The calculator uses JavaScript's built-in math functions and displays results to 4 decimal places, providing high precision for most practical applications.
What are some real-world applications?
Heron's formula is used in land surveying to calculate plot areas, in navigation to determine distances, in architecture for irregular triangular spaces, and in computer graphics for polygon rendering.
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