45 45 90 Triangle Calculator
Calculate the sides and area of a 45-45-90 special right triangle. Enter either a leg length or the hypotenuse, and this calculator finds all remaining measurements using the ratio 1 : 1 : √2.
Properties of the 45-45-90 Triangle
The 45-45-90 triangle is one of two special right triangles that appear frequently in geometry and trigonometry. Because its two acute angles are equal at 45 degrees each, it is also an isosceles triangle, meaning both legs have the same length. The side ratio 1 : 1 : √2 makes calculations straightforward once you know any single side.
This triangle emerges naturally when you bisect a square along its diagonal. The diagonal becomes the hypotenuse, and the two sides of the square become the legs. This geometric relationship is why √2 appears so often in construction and design calculations involving squares and rectangles. Understanding this triangle simplifies many practical measurement problems.
Trigonometric Values at 45 Degrees
The 45-45-90 triangle provides the exact trigonometric values for 45 degrees. Since the two legs are equal, sin(45°) = cos(45°) = leg/hypotenuse = 1/√2 = √2/2 ≈ 0.7071. The tangent of 45° equals 1 because the opposite and adjacent sides are the same length.
These values are essential in trigonometry, physics, and engineering. Knowing that sin(45°) = √2/2 without a calculator is expected on standardized tests like the SAT and ACT. The exact values also appear in rotation matrices, vector decomposition, and wave analysis. Memorizing the 45-45-90 ratio eliminates the need to derive these trigonometric values from scratch each time.
Real-World Applications
In construction, the 45-45-90 triangle is used to verify square corners and calculate diagonal bracing lengths. If a square room measures 10 feet on each side, the diagonal measures 10√2 ≈ 14.14 feet. This knowledge helps carpenters, architects, and engineers plan layouts and verify accuracy during building projects.
In navigation, a 45-degree heading change creates this triangle pattern. Pilots and sailors use it when computing course corrections. In computer graphics, 45-degree rotations rely on the √2 ratio for pixel-accurate rendering. Even in everyday life, knowing that a diagonal shortcut across a square field is about 1.41 times the side length helps estimate walking distances and plan efficient routes.
Frequently Asked Questions
What is a 45-45-90 triangle?
A 45-45-90 triangle is an isosceles right triangle where the two legs are equal in length and the angles measure 45°, 45°, and 90°. The sides follow the ratio 1 : 1 : √2.
How do you find the hypotenuse from a leg?
Multiply the leg length by √2 (approximately 1.4142). If the leg is 5, the hypotenuse is 5√2 ≈ 7.0711.
How do you find the leg from the hypotenuse?
Divide the hypotenuse by √2 or equivalently multiply by √2/2. If the hypotenuse is 10, each leg is 10/√2 ≈ 7.0711.
Where do 45-45-90 triangles appear in real life?
They appear when you cut a square along its diagonal, in roof framing, in engineering designs, and whenever equal sides meet at a right angle.
What is the area formula for a 45-45-90 triangle?
The area is (1/2) × leg × leg, or equivalently leg²/2, since both legs are equal. If the leg is a, the area is a²/2.