45-45-90 Triangle Calculator
Calculate legs, hypotenuse, area, and perimeter of special 45-45-90 isosceles right triangles
45-45-90 Triangle Calculator
Both legs of a 45-45-90 triangle are equal in length
45-45-90 Triangle Diagram
Example Calculation
Drafting Set Square Example
Given: Leg length = 9 inches
Type: 45-45-90 triangle (isosceles right triangle)
This is commonly found in drafting set squares and technical drawing tools.
Solution
Both legs: a = b = 9 inches (given)
Hypotenuse: c = a√2 = 9√2 ≈ 12.73 inches
Area: A = a²/2 = 9²/2 = 40.5 square inches
Perimeter: P = a(2 + √2) = 9(2 + √2) ≈ 30.73 inches
Verification: P = 9 + 9 + 12.73 = 30.73 inches ✓
45-45-90 Triangle Properties
Isosceles Right Triangle
Angles: 45°, 45°, 90°
Two equal legs, one right angle
Equal Legs
Both legs: a = b
Hypotenuse: c = a√2
Half of a Square
Diagonal cuts square in half
Diagonal becomes hypotenuse
Formula Reference
Side Ratios
a : a : a√2
From Leg (a)
c = a√2
From Hypotenuse (c)
a = c/√2
Area
A = a²/2
Perimeter
P = a(2 + √2)
From Area
a = √(2A)
From Perimeter
a = P/(2 + √2)
Understanding 45-45-90 Triangles
What Makes This Triangle Special?
A 45-45-90 triangle is a special right triangle with two 45° angles and one 90° angle. It's the only right triangle that is also isosceles, meaning two sides (the legs) are equal in length. This makes it exactly half of a square cut along its diagonal.
Unique Properties
- •Only right triangle that is also isosceles
- •Side ratio is always 1 : 1 : √2
- •Smallest hypotenuse-to-legs ratio of all right triangles
- •Exactly half of a square
Real-World Applications
- •Drafting and technical drawing set squares
- •Construction and carpentry (roof slopes)
- •Architecture and structural design
- •Engineering and mechanical design
- •Computer graphics and game development
Memory Tip: Think of it as "half a square." If you cut a square diagonally, you get two 45-45-90 triangles. The diagonal becomes the hypotenuse, and its length is √2 times the side of the original square.
Mathematical Derivation
Pythagorean Theorem
Given: a² + b² = c²
Since a = b in 45-45-90 triangle:
a² + a² = c²
2a² = c²
c = a√2
Square Diagonal
Square with side length a
Diagonal = side × √2
Diagonal = a√2
Half-square triangle has hypotenuse = a√2
Trigonometry
sin(45°) = cos(45°) = √2/2
a/c = sin(45°) = √2/2
Therefore: c = a/(√2/2) = a√2
This confirms c = a√2