Isosceles Right Triangle Calculator
Calculate area, perimeter, height, and side lengths of isosceles right triangles (45-45-90 triangles)
Calculate Isosceles Right Triangle
Length of the two equal sides (legs) of the triangle
Triangle Properties
Triangle Properties
Example Calculation
Example: Square with diagonal
Problem: A square has a diagonal of 10 cm. Find the side length.
Solution: The diagonal creates two isosceles right triangles
Given: Hypotenuse (diagonal) = 10 cm
Side length: a = h/√2 = 10/√2 = 7.07 cm
Verification
Using Pythagorean theorem: a² + a² = h²
7.07² + 7.07² = 50 + 50 = 100
√100 = 10 cm ✓
Key Formulas
Pythagorean Theorem
h² = a² + a² = 2a²
h = a√2
Area
A = ½ × a × a = a²/2
Perimeter
P = 2a + h = a(2 + √2)
Angles
45° - 45° - 90°
45-45-90 Triangle
Also called isosceles right triangle
Two equal sides (legs) and equal base angles
Ratio of sides is 1:1:√2
Appears in squares cut diagonally
Common in architecture and design
Quick Tips
Memorize the ratio 1:1:√2 for quick calculations
√2 ≈ 1.414 for approximations
Always check: a² + a² = h²
Useful for unit circle calculations
Understanding Isosceles Right Triangles
What is an Isosceles Right Triangle?
An isosceles right triangle is a special type of triangle that combines the properties of both isosceles triangles (two equal sides) and right triangles (one 90° angle). It's also known as a 45-45-90 triangle because of its three angles.
Key Properties
- •Two equal sides (legs) that meet at a right angle
- •Two equal base angles of 45° each
- •One right angle (90°) opposite the hypotenuse
- •Side ratio of 1:1:√2 (leg:leg:hypotenuse)
Practical Applications
Architecture & Construction
Used in roof designs, diagonal bracing, and creating stable structural elements.
Engineering
Common in mechanical designs, force analysis, and geometric calculations.
Mathematics
Foundation for trigonometry, coordinate geometry, and vector mathematics.
Formula Derivations
From Equal Sides to Hypotenuse
Given: Equal sides = a
Using Pythagorean theorem:
h² = a² + a²
h² = 2a²
h = a√2
From Hypotenuse to Equal Sides
Given: Hypotenuse = h
Using h = a√2:
a√2 = h
a = h/√2
a = h√2/2