45 45 90 Triangle Calculator: How It Works
The 45-45-90 triangle is one of the most important special right triangles in geometry. It is an isosceles right triangle — meaning two legs are equal in length and the two acute angles are each exactly 45°. This unique shape appears everywhere from architecture to engineering to everyday set squares.
45 45 90 Triangle Formulas
If the leg length is a, then:
- Second leg: b = a (both legs are equal)
- Hypotenuse: c = a√2 ≈ a × 1.4142
- Area: A = a² / 2
- Perimeter: P = a(2 + √2) ≈ a × 3.4142
How to Solve a 45 45 90 Triangle
Depending on what you know, you can solve the triangle in three ways:
- From leg (a): hypotenuse = a√2, area = a²/2, perimeter = a(2+√2)
- From hypotenuse (c): leg = c/√2 = c√2/2, area = c²/4, perimeter = c(1+√2)
- From area (A): leg = √(2A), hypotenuse = 2√A, perimeter = √(2A)×(2+√2)
45 45 90 Triangle Rules
- One right angle (90°) and two equal angles (45° each)
- The two legs are always equal in length
- The hypotenuse is always √2 times the length of a leg
- It is the only right triangle that is also isosceles
- The side ratio is always 1 : 1 : √2
45 45 90 Triangle Ratio
The ratios for this triangle are:
- Angles: 45° : 45° : 90°
- Sides: a : a : a√2 (or 1 : 1 : √2)
Example: Solving a 45 45 90 Triangle
If the leg equals 9 inches (US customary):
- Hypotenuse = 9 × √2 ≈ 12.73 in
- Area = 9² / 2 = 81 / 2 = 40.5 in²
- Perimeter = 9 + 9 + 12.73 ≈ 30.73 in
The same leg in metric (cm): leg = 22.86 cm → hypotenuse ≈ 32.32 cm → area ≈ 261.29 cm².
Unit Systems
Our calculator supports both:
- Metric system: millimeters (mm), centimeters (cm), meters (m), kilometers (km)
- US Customary (Imperial): inches (in), feet (ft), yards (yd), miles (mi)
Why is the 45 45 90 Triangle Special?
- It is half of a square, cut along the diagonal
- Of all right triangles, it has the smallest ratio of hypotenuse to sum of legs
- Of all right triangles, it has the greatest altitude ratio from hypotenuse to sum of legs
- It appears in set squares, tile patterns, and architectural designs
Frequently Asked Questions
Q: What are the sides of a 45 45 90 triangle?
A: The two legs are equal (both = a), and the hypotenuse equals a√2.
Q: How do I find the hypotenuse if I know the leg?
A: Multiply the leg by √2 (≈ 1.41421).
Q: How do I find the leg if I know the hypotenuse?
A: Divide the hypotenuse by √2, or multiply by √2/2 (≈ 0.70711).
Q: Can I use this calculator with inches and feet?
A: Yes! Select "Imperial" as your unit system to work with US customary units.