45-45-90 Triangle
Sides by hypotenuse
45-45-90 Triangle Calculator
Unlock the simplicity of special right triangles with our 45-45-90 triangle calculator. This tool is perfect for students, engineers, and math enthusiasts needing quick calculations for isosceles right triangles. Whether you’re solving for the hypotenuse, legs, or area, our calculator provides accurate results instantly. Optimize your geometry tasks with this SEO-optimized guide and tool.
About
A 45-45-90 triangle is a special type of right-angled triangle where two angles are 45 degrees each, and the third is 90 degrees. It’s also isosceles, meaning the two legs are equal in length. The key property is the side ratio: if each leg is of length x, the hypotenuse is x√2. This ratio derives from the Pythagorean theorem: hypotenuse = √(x² + x²) = √(2x²) = x√2.
Our 45-45-90 triangle calculator simplifies these computations. Input any known side—leg or hypotenuse—and it computes the unknowns, including perimeter and area. It’s essential for trigonometry, architecture, and DIY projects involving 45-degree angles. By using this calculator, you save time and avoid manual errors, making it a go-to resource for precise geometric solutions.
How to Use
Using our 45-45-90 triangle calculator is straightforward. Start by selecting the known value: either a leg length or the hypotenuse. Enter the numerical value in the input field. For example, if you know one leg is 5 units, input 5 under “Leg Length.” Hit the “Calculate” button, and the tool will display the other leg (also 5), hypotenuse (5√2 ≈ 7.07), area (12.5), and perimeter (≈17.07).
If you have the hypotenuse, say 10, input it, and the calculator finds each leg as 10/√2 ≈ 7.07. Remember to use decimal approximations for practical applications, but the tool provides exact values with √2. No advanced math skills needed—just plug in and get results. This user-friendly interface ensures accessibility for all levels.
Examples
Example 1: Suppose you have a 45-45-90 triangle with a leg of 3 cm. Using the calculator, the other leg is 3 cm, hypotenuse is 3√2 ≈ 4.24 cm, area is (3*3)/2 = 4.5 cm², perimeter is 3 + 3 + 4.24 ≈ 10.24 cm.
Example 2: For a hypotenuse of 8 units, each leg is 8/√2 ≈ 5.66 units. Area: (5.66*5.66)/2 ≈ 16 units². This is useful in roofing where diagonal supports form such triangles.
Example 3: In a square with side 4, the diagonal forms two 45-45-90 triangles. Calculator confirms diagonal as 4√2 ≈ 5.66.
FAQ
1. What is the ratio in a 45-45-90 triangle? The sides are in the ratio 1 : 1 : √2, where the legs are equal, and the hypotenuse is leg times √2.
2. How do I find the area of a 45-45-90 triangle? Area = (leg²)/2. Our calculator computes it automatically from any input.
3. Can this calculator handle decimal inputs? Yes, it accepts decimals and fractions, providing both exact and approximate outputs.
4. Is a 45-45-90 triangle always isosceles? Yes, by definition, the two legs are congruent.
5. What real-world applications use 45-45-90 triangles? They’re common in construction, graphic design, and physics for calculating forces at 45-degree angles.