45-45-90 Triangle Calculator
Isosceles right triangle with sides in ratio 1 : 1 : √2
Enter Any Side
Side Ratios
1 : 1 : √2
• Both legs are equal = x
• Hypotenuse = x√2
• Leg from hypotenuse = hyp/√2 = hyp√2/2
Also Known As
- • Isosceles Right Triangle
- • Half of a Square
45-45-90 Triangle
Understanding the 45-45-90 Triangle
The 45-45-90 triangle is a special right triangle also known as an isosceles right triangle. It has two equal angles of 45° each and one right angle of 90°. The two legs are always equal in length.
Origin: Half of a Square
A 45-45-90 triangle is exactly half of a square, created by drawing a diagonal from one corner to the opposite corner. This is why the two legs are equal - they're sides of the original square.
The Side Relationships
- Both legs (x): Equal in length, opposite the 45° angles
- Hypotenuse (x√2): Opposite the 90° angle, the diagonal of a square
Conversion Formulas
- If leg = x: hypotenuse = x√2
- If hypotenuse = h: leg = h/√2 = h√2/2
Why √2?
By the Pythagorean theorem: if both legs equal x, then hypotenuse² = x² + x² = 2x². Taking the square root gives hypotenuse = x√2.
Common Applications
- Finding diagonal of a square
- Calculating distances at 45° angles
- Architecture and construction
- Computer graphics and game development
Frequently Asked Questions
Why is the ratio 1 : 1 : √2?
From the Pythagorean theorem: if both legs = 1, then hypotenuse = √(1² + 1²) = √2. The two legs are equal because it's isosceles.
Why is this called an isosceles right triangle?
It's isosceles because two sides (the legs) are equal. It's right because it has a 90° angle. The 45-45-90 name comes from its angle measures.
How is this related to a square?
A 45-45-90 triangle is exactly half of a square. The legs are sides of the square, and the hypotenuse is the diagonal of the square.
What is √2 approximately equal to?
√2 ≈ 1.414. So if each leg is 1, the hypotenuse is about 1.414.
How do I rationalize √2 in the denominator?
Multiply by √2/√2. For example, h/√2 = h√2/2. This gives an equivalent form without a radical in the denominator.
What are the exact trig values for 45°?
sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1. Notice sin and cos are equal because the triangle is isosceles.
What's the area formula for a 45-45-90 triangle?
Area = (1/2) × leg × leg = leg²/2. Since both legs are equal, you just square the leg and divide by 2.
When should I use this vs the Pythagorean theorem?
Use the 1:1:√2 ratio whenever you know it's a 45-45-90 triangle. It's faster than calculating with the Pythagorean theorem each time.