45-45-90 Triangle Calculator
Solve a 45-45-90 special right triangle from any one known side. Get the leg, hypotenuse, area, and perimeter using the 1:1:sqrt2 ratio instantly with our free online geometry tool.
About This Calculator
This free online 45-45-90 triangle calculator solves the triangle from any one known side. Select whether you know the leg or the hypotenuse, enter its length, and the calculator uses the fixed side ratio of 1 : 1 : sqrt2 to find all remaining dimensions including area and perimeter.
A 45-45-90 triangle, also called an isosceles right triangle, has two equal legs and angles of 45 deg, 45 deg, and 90 deg. Since both legs are equal, the side ratio is straightforward: leg : leg : hypotenuse = 1 : 1 : sqrt2. This makes it one of the most commonly encountered special right triangles in geometry, construction, engineering, and trigonometry.
The side lengths follow directly from the Pythagorean theorem: a^2 + a^2 = c^2, which simplifies to c = asqrt2. As a result, if you know the leg length, multiply it by sqrt2 to get the hypotenuse. If you know the hypotenuse instead, divide by sqrt2 to get each leg. The calculator also computes the area as half the square of the leg length (a^2/2) and the perimeter as 2a + asqrt2.
45-45-90 triangles appear in many real-world contexts. In construction and carpentry, they are used for roof trusses, stair stringers, and miter cuts. In digital graphics and game development, the 45-45-90 triangle is fundamental for calculating diagonal distances on pixel grids. In navigation and surveying, the isosceles right triangle simplifies distance calculations when turning 90 degrees at equal intervals.
Whether you are a student preparing for geometry exams or a professional needing quick triangle dimensions, this tool provides instant accurate results with a clean, easy-to-use interface.
Frequently Asked Questions
What is a 45-45-90 triangle?
A 45-45-90 triangle is a special right triangle whose angles are 45 deg, 45 deg, and 90 deg. It is also called an isosceles right triangle. The side lengths follow the fixed ratio 1 : 1 : sqrt2, where both legs are equal and the hypotenuse is opposite the 90 deg angle.
What are the side ratios of a 45-45-90 triangle?
In a 45-45-90 triangle, both legs are equal. If each leg is x, then the hypotenuse is xsqrt2. The side ratio is always 1 : 1 : sqrt2.
How do you find the sides of a 45-45-90 triangle?
Select whether you know the leg or the hypotenuse, then enter its length. If you know the leg, multiply by sqrt2 to get the hypotenuse. If you know the hypotenuse, divide by sqrt2 to get the leg length.
How do you find the area of a 45-45-90 triangle?
The area of a 45-45-90 triangle is calculated as 1/2 x leg x leg = leg^2/2. For example, if the leg is 6, the area is 6^2/2 = 18 square units.
How do you find the perimeter of a 45-45-90 triangle?
Add all three sides: leg + leg + hypotenuse = 2a + asqrt2 = a(2 + sqrt2). For example, if the leg is 6, the perimeter is 6 + 6 + 6sqrt2 ≈ 20.49 units.
What is the difference between a 45-45-90 and a 30-60-90 triangle?
A 45-45-90 triangle has equal legs and angles of 45-45-90, with ratio 1:1:sqrt2. A 30-60-90 triangle has legs in the ratio 1:sqrt3 and angles 30-60-90, with ratio 1:sqrt3:2. Both are special right triangles but with different angle and side proportions.
What are the real-world applications of 45-45-90 triangles?
45-45-90 triangles appear in construction (roof trusses, set squares), engineering (bridge trusses), computer graphics (pixel-based diagonal lines), and navigation. The isosceles right triangle's predictable ratio makes it ideal for quick geometric calculations in design and architecture.
Is this calculator free to use?
Yes, all calculators on Calculy are completely free to use with no registration required.