45-45-90 Triangle Calculator - Isosceles Right Triangle
Find all sides of a 45-45-90 isosceles right triangle instantly by entering any one known side length.
Select whether you know a leg or the hypotenuse, enter its length, and the calculator applies the exact 1 : 1 : √2 ratio to find all sides.
45-45-90 Triangle Calculator - Isosceles Right Triangle
Find all sides of a 45-45-90 isosceles right triangle instantly by entering any one known side length.
About the 45-45-90 Triangle Calculator
A 45-45-90 triangle is the second of the two most important special right triangles in mathematics, alongside the 30-60-90 triangle. Named for its three interior angles of 45°, 45°, and 90°, this triangle is also called an isosceles right triangle because its two legs are always equal in length. The sides follow the exact ratio 1 : 1 : √2, where each leg has length 1 unit and the hypotenuse has length √2 ≈ 1.41421 units. As with the 30-60-90 triangle, this ratio scales perfectly: knowing any one side immediately determines all three sides.
The 45-45-90 triangle arises from cutting a square along its diagonal. A square with side length 1 produces two congruent 45-45-90 triangles whose legs are the square's sides and whose hypotenuse is the square's diagonal. By the Pythagorean theorem, that diagonal equals √(1² + 1²) = √2. This simple construction confirms why the ratio is exact and why √2 — despite being irrational — is a fundamental constant in geometry.
To find the hypotenuse from a known leg, multiply by √2 (approximately 1.41421356). To find a leg from the hypotenuse, divide by √2, equivalently multiply by √2/2 ≈ 0.70711. Because both legs are equal, knowing either leg gives the other immediately. The calculator performs these operations in full double-precision floating-point arithmetic, giving results accurate to at least 10 significant digits.
In architecture and design, 45-45-90 triangles are ubiquitous. Roof trusses using a 45-degree pitch, corner bracing in wooden frames, and mitre cuts at 45 degrees all rely on this triangle. Tile layouts often use 45-degree diagonals, where understanding the relationship between tile size and diagonal length prevents wasteful cuts. In computer graphics and game development, 45-degree angles appear constantly in pixel art grids, isometric projections, and collision geometry.
The trigonometric values for 45 degrees — sin(45°) = cos(45°) = √2/2 — come directly from this triangle and appear throughout calculus, signal processing, and physics. In physics, the 45-degree launch angle maximises projectile range on a flat surface; the horizontal and vertical components of velocity are equal at that angle, which is the direct consequence of a 45-45-90 triangle decomposition. In woodworking, a perfect mitre joint requires that two pieces be cut at exactly 45 degrees, and the diagonal of the resulting square corner is the hypotenuse of a 45-45-90 triangle.
Everywhere that equal proportions and right angles combine — from picture frames to window sashes to the cuts made by a table saw at 45 degrees — the 45-45-90 triangle provides the governing geometry. This calculator automates the arithmetic so you can focus on the design or engineering task at hand.
45-45-90 triangle examples
These examples show how the 1 : 1 : √2 ratio lets you find all sides from any single known measurement.
| Known Side | All Three Sides | Explanation |
|---|---|---|
| Leg = 5 | Leg 1 = 5, Leg 2 = 5, Hypotenuse ≈ 7.071 | Multiply the leg by √2 ≈ 1.41421 to get the hypotenuse. Both legs are equal, so the triangle is isosceles. This is the geometry of a 5-unit square cut diagonally. |
| Hypotenuse = 10 | Leg 1 ≈ 7.071, Leg 2 ≈ 7.071, Hypotenuse = 10 | Divide the hypotenuse by √2 to find each leg. Equivalently, multiply by √2/2 ≈ 0.70711. This configuration appears when designing an equal-leg corner brace for a 10-unit span. |
| Leg = 1 | Leg 1 = 1, Leg 2 = 1, Hypotenuse = √2 ≈ 1.41421 | The unit 45-45-90 triangle is the canonical form of this special triangle. Its hypotenuse is exactly √2, an irrational number. This is the diagonal of a unit square. |
How to use the 45-45-90 Triangle Calculator
- Select which side you know: choose Leg (equal sides) if you know one of the two equal legs, or Hypotenuse (opposite 90°) if you know the longest side.
- Enter the length of the known side as a positive number in the Side Length field.
- Click Calculate Triangle to instantly see all three sides.
- Confirm the result: the two legs should be equal, and the hypotenuse should be the leg multiplied by √2 ≈ 1.41421.
- Click Reset to clear the inputs, or switch the side type and re-enter a value to verify your answer from a different direction.
45-45-90 Triangle FAQ
What is the side ratio of a 45-45-90 triangle?
The sides are in the ratio 1 : 1 : √2. The two legs are always equal, and the hypotenuse is exactly √2 times the length of either leg. This ratio is mathematically exact and scales to any size of 45-45-90 triangle.
How do I find the hypotenuse if I know a leg?
Multiply the leg by √2, which is approximately 1.41421356. For example, a leg of 6 gives a hypotenuse of 6√2 ≈ 8.485. The calculator provides this multiplication to full floating-point precision.
How do I find a leg if I know the hypotenuse?
Divide the hypotenuse by √2, or equivalently multiply by √2/2 ≈ 0.70711. For a hypotenuse of 14, each leg is 14/√2 = 7√2 ≈ 9.899. Because both legs are equal, this single calculation gives both legs at once.
Why are the two legs always equal?
The two legs are equal because the triangle is isosceles — the 45° angles at each base are equal, so the sides opposite those equal angles are also equal. This is also why the triangle is sometimes called an isosceles right triangle.
Where do 45-45-90 triangles appear in real life?
They appear whenever a square is cut diagonally, including in roof framing at 45-degree pitches, mitre-joint woodworking, corner bracing, tile diagonal layouts, and pixel-art grids. In physics, the 45-degree launch angle for maximum projectile range is governed by this triangle's equal horizontal and vertical velocity components.
What are sin, cos, and tan of 45 degrees?
From the 45-45-90 triangle with legs of length 1 and hypotenuse √2: sin(45°) = 1/√2 = √2/2 ≈ 0.7071, cos(45°) = 1/√2 = √2/2 ≈ 0.7071, and tan(45°) = 1/1 = 1. These are exact values used throughout trigonometry and calculus.