Heron's Formula Calculator
Calculate triangle area using Heron's formula when three side lengths are known
Calculate Triangle Area
First side length
Second side length
Third side length
Classic Example: 3-4-5 Triangle
Problem
Find the area of a triangle with sides 3, 4, and 5 units using Heron's formula.
Solution
Step 1: Calculate semiperimeter
s = (3 + 4 + 5) / 2 = 12 / 2 = 6
Step 2: Apply Heron's formula
A = √(s(s-a)(s-b)(s-c))
A = √(6 × (6-3) × (6-4) × (6-5))
A = √(6 × 3 × 2 × 1)
A = √36 = 6 units²
Note: This is a right triangle (3² + 4² = 5²), so we can verify: A = ½ × 3 × 4 = 6 ✓
Heron's Formula
Main Formula
Where s is the semiperimeter
Semiperimeter
Half of the triangle's perimeter
Alternative Form
More numerically stable for thin triangles
Triangle Inequality
For a valid triangle, the sum of any two sides must be greater than the third side:
If any condition fails, the three lengths cannot form a triangle.
Understanding Heron's Formula
What is Heron's Formula?
Heron's formula (also known as Hero's formula) is a mathematical formula that calculates the area of a triangle when you know the lengths of all three sides. It was first mentioned in Heron's book "Metrica" around 60 AD.
When to Use It
- •When you know all three side lengths
- •When the triangle's height is unknown
- •For any type of triangle (scalene, isosceles, equilateral)
- •In surveying and engineering applications
Formula Derivation
Algebraic Proof
1. Start with basic area formula: A = (base × height) / 2
2. Use Pythagorean theorem to express height in terms of sides
3. Substitute and simplify using algebraic identities
4. Result: A = √(s(s-a)(s-b)(s-c))
Trigonometric Proof
1. Start with: A = ½ab sin(C)
2. Use law of cosines: c² = a² + b² - 2ab cos(C)
3. Apply identity: sin²(C) + cos²(C) = 1
4. Substitute and simplify to get Heron's formula
Advantages and Limitations
Advantages
- • Works with any triangle type
- • Only requires side lengths
- • No need to know angles or heights
- • Exact result (no approximation)
Limitations
- • Can be numerically unstable for very thin triangles
- • Requires all three sides to be known
- • Square root calculation needed
- • Must verify triangle inequality first