Isosceles Triangle Area Calculator
Calculate triangle area using multiple methods with step-by-step solutions
Calculate Isosceles Triangle Area
Length of the two equal sides
Length of the third side
Area Calculation Results
Enter the required values to calculate the triangle area.
Example Calculation
Example: Garden Triangle Design
Problem: A triangular garden has two equal sides of 13m each and a base of 24m. What is the area?
Given: Leg length (a) = 13m, Base length (b) = 24m
Method: Using legs and base (most common)
Solution
Step 1: Check triangle validity: 2(13) = 26 > 24 ✓
Step 2: Calculate height using Pythagorean theorem
Formula: h = √[a² - (b/2)²] = √[13² - (24/2)²]
Calculate: h = √[169 - 144] = √25 = 5m
Step 3: Calculate area
Formula: A = ½ × b × h = ½ × 24 × 5 = 60m²
Result: The garden area is 60 square meters
Area Formulas
Standard Formula
A = ½ × base × height
Height from Legs & Base
h = √[a² - (b/2)²]
From Legs & Angle
A = a² × sin(α) × cos(α)
Heron's Formula
A = √[s(s-a)(s-a)(s-b)]
where s = (2a + b)/2
Method Comparison
Legs + Base
Most common, uses Pythagorean theorem
Base + Height
Direct calculation, fastest method
Legs + Angle
Uses trigonometry, good for engineering
Reverse Calculate
Find dimensions from known area
Triangle Properties
Two equal sides (legs)
Two equal base angles
Height bisects base and vertex angle
Axis of symmetry through vertex
Triangle inequality: 2a > b
Understanding Isosceles Triangle Area Calculation
What Makes This Calculator Special?
This calculator offers five different methods to calculate isosceles triangle area, making it versatile for various scenarios. It handles forward calculations (dimensions to area) and reverse calculations (area to dimensions).
Key Mathematical Relationships
- •Pythagorean relationship: h² + (b/2)² = a²
- •Area formula: A = ½ × base × height
- •Trigonometric area: A = a² × sin(α) × cos(α)
- •Height from angle: h = a × sin(α)
Calculation Methods
Most Common: Legs + Base
Uses the Pythagorean theorem to find height, then applies the standard area formula. Perfect when you know the side lengths.
Simplest: Base + Height
Direct application of A = ½bh. Use this when the height is already known or measured.
Engineering: Legs + Angle
Uses trigonometry to find both base and height from the leg length and base angle. Common in engineering applications.
Real-World Applications
Architecture & Construction
Calculate roof areas, triangular supports, and architectural features with isosceles geometry.
Landscaping
Design triangular gardens, calculate lawn areas, and plan irrigation coverage for triangular plots.
Education & Research
Solve geometry problems, understand mathematical relationships, and verify homework solutions.