Isosceles Right Triangle Hypotenuse Calculator
Calculate the hypotenuse length of 45-45-90 triangles from equal side lengths
Calculate Hypotenuse Length
Length of the two equal sides (legs) of the isosceles right triangle
Hypotenuse Result
Mathematical Properties
Example Calculation
Example: Square Diagonal Problem
Problem: A square has sides of 5 cm each. What is the length of its diagonal?
Solution: The diagonal divides the square into two isosceles right triangles
Given: Equal sides (B) = 5 cm
Formula: A = B × √2
Calculation: A = 5 × √2 = 5 × 1.414 = 7.071 cm
Verification using Pythagorean Theorem
A² = B² + B²
A² = 5² + 5² = 25 + 25 = 50
A = √50 = √(25 × 2) = 5√2 ≈ 7.071 cm ✓
Hypotenuse Formula
A = Hypotenuse length
B = Equal side length
√2 ≈ 1.414
Pythagorean Theorem Form
A² = B² + B² = 2B²
A = √(2B²) = B√2
Triangle Properties
Two equal sides (legs)
One right angle (90°)
Two equal base angles (45° each)
Side ratio 1:1:√2
Also called 45-45-90 triangle
Common Applications
Square diagonal calculations
Architectural measurements
Engineering designs
Trigonometry problems
Construction layouts
Understanding Isosceles Right Triangle Hypotenuse
What Makes It Special?
An isosceles right triangle is unique because it combines the properties of both isosceles triangles (two equal sides) and right triangles (one 90° angle). This creates a triangle with very predictable and useful mathematical relationships.
The Hypotenuse Formula
The hypotenuse of an isosceles right triangle is always √2 times the length of either equal side. This comes directly from the Pythagorean theorem: A² = B² + B² = 2B², so A = B√2.
Why √2?
- •√2 ≈ 1.414 is an irrational number
- •It represents the diagonal of a unit square
- •Fundamental constant in geometry
Step-by-Step Derivation
Step 1: Apply Pythagorean Theorem
A² = B² + B²
Step 2: Simplify
A² = 2B²
Step 3: Take Square Root
A = √(2B²) = B√2
Final Formula
A = B × √2
Real-World Applications
Construction
Calculating diagonal braces, corner cuts, and structural supports in buildings.
Design
Creating square layouts, determining screen diagonals, and graphic design proportions.
Mathematics
Foundation for trigonometry, coordinate geometry, and advanced mathematical concepts.