3 Sides Triangle Area Calculator
Calculate triangle area using Heron's formula when you know all three side lengths
Calculate Triangle Area
Length of the first side
Length of the second side
Length of the third side
Triangle Area Results
Example Calculation
Example: Triangle with sides 9, 6, and 5 inches
Given: a = 9 in, b = 6 in, c = 5 in
Check validity: 9 + 6 > 5 ✓, 9 + 5 > 6 ✓, 6 + 5 > 9 ✓
Semi-perimeter: s = (9 + 6 + 5) / 2 = 10 in
Step-by-Step Solution
A = √[s(s-a)(s-b)(s-c)]
A = √[10 × (10-9) × (10-6) × (10-5)]
A = √[10 × 1 × 4 × 5]
A = √200
A = 14.14 square inches
Heron's Formula
A = √[s(s-a)(s-b)(s-c)]
Where s = (a+b+c)/2
A: Area of the triangle
a, b, c: Lengths of the three sides
s: Semi-perimeter of the triangle
Triangle Inequality Theorem
a + b > c
a + c > b
b + c > a
Note: All three conditions must be true for a valid triangle.
Triangle Types
Equilateral
All sides are equal
Isosceles
Two sides are equal
Scalene
All sides are different
Right
Has a 90° angle (a² + b² = c²)
Understanding Heron's Formula
What is Heron's Formula?
Heron's formula, discovered by Hero of Alexandria around 60 AD, allows us to calculate the area of any triangle when we know the lengths of all three sides. It's particularly useful when we don't know the height of the triangle.
Why is it Important?
- •Works with any triangle (scalene, isosceles, equilateral)
- •No need to know angles or height
- •Useful in surveying and construction
- •Foundation for more complex geometric calculations
Formula Derivation
A = √[s(s-a)(s-b)(s-c)]
Where s = (a+b+c)/2
Step 1: Calculate the semi-perimeter (s)
Step 2: Subtract each side from the semi-perimeter
Step 3: Multiply s with all three differences
Step 4: Take the square root of the result
Alternative Formula: A = ¼√[4a²b² - (a² + b² - c²)²]
Real-World Applications
🏗️ Construction & Architecture
Calculate areas of triangular plots, roof sections, and structural elements.
📐 Surveying & Mapping
Determine land areas and property boundaries using triangulation methods.
🎨 Design & Graphics
Calculate areas in graphic design, fabric cutting, and artistic layouts.
⛵ Navigation
Calculate distances and areas using GPS coordinates and triangulation.
🏡 Home Improvement
Calculate material needs for triangular spaces, gardens, and room layouts.
🔬 Engineering
Analyze stress distributions, force calculations, and structural design.