45-45-90 Triangle Calculator
Calculate the leg, hypotenuse, area, and perimeter of any 45-45-90 right triangle.
What is a 45-45-90 Triangle?
A 45-45-90 triangle is a special right triangle with interior angles of exactly 45 degrees, 45 degrees, and 90 degrees. Because both acute angles are equal, the two legs opposite them are also equal, making it an isosceles right triangle. This combination of right angle and isosceles property gives the 45-45-90 triangle its unique and predictable behaviour: the sides are always in the fixed ratio 1 : 1 : √2, no matter what size the triangle is.
This means that if you know any one side, you can calculate every other dimension without needing any trigonometric tables. The hypotenuse is always exactly √2 (approximately 1.41421) times each leg, and the two legs are always equal to each other. This predictability makes the 45-45-90 triangle one of the most commonly encountered shapes in geometry, architecture, engineering, and everyday design work.
The 45-45-90 triangle appears naturally whenever a square is divided diagonally. If a square has side length s, its diagonal measures s√2, and the two triangles formed are both 45-45-90 right triangles with legs equal to s and hypotenuse equal to s√2. This is also the origin of the √2 factor in the diagonal formula for a square.
In trigonometry, the 45-degree angle is fundamental because sin(45°) = cos(45°) = √2/2 ≈ 0.7071, and tan(45°) = 1. These are among the most frequently used exact trigonometric values, derived directly from the 1 : 1 : √2 ratio of this triangle. Students who memorise the 45-45-90 ratios can solve many geometry and trigonometry problems mentally, without a calculator.
Practically, 45-degree angles appear everywhere: roof pitch calculations, mitre cuts in woodworking, diagonal bracing in structural engineering, and perspective drawing in art all rely on 45-degree geometry. The drafting set square is a physical 45-45-90 triangle used by architects and engineers daily.
Formula and Derivation
The 45-45-90 triangle formulas follow directly from the Pythagorean theorem applied to an isosceles right triangle.
Given leg a:
Given hypotenuse h:
Derivation: For a right triangle with two equal legs a, the Pythagorean theorem gives h² = a² + a² = 2a², so h = a√2. Solving for a: a = h/√2 = h√2/2.
How to Use This Calculator
- Choose your known measurement - click “Enter Leg” if you know the length of one leg, or “Enter Hypotenuse” if you know the hypotenuse. Both legs in a 45-45-90 triangle are always equal, so entering either leg gives the same result.
- Type the value - enter any positive number. You can use any unit of length (cm, m, inches, feet). All results will be in the same unit, and area will be in square units.
- Click Calculate - press the button to instantly compute all four measurements: each leg, hypotenuse, area, and perimeter.
- Check the formula note - below the results you’ll see the exact formula applied to your input, which you can use to verify by hand.
- Try an example link - the worked examples below each include a “Try this example” link that pre-fills the calculator with real values.
Example Calculations
Example 1 - Leg of 5 units
Find the hypotenuse, area, and perimeter of a 45-45-90 triangle with legs of 5 cm each.
Example 2 - Hypotenuse of 10 units
A hypotenuse of 10 m is given. Find the legs, area, and perimeter.
Example 3 - Leg of 8 units (tile diagonal problem)
A square ceramic tile has a side of 8 inches. A cut along the diagonal creates two 45-45-90 triangles. Find all dimensions of each triangle.
Example 4 - Hypotenuse of 14.142 units
A ramp forms a 45-degree angle and has a slant length (hypotenuse) of 14.142 m. Find the horizontal run and vertical rise.