45-45-90 Triangle Calculator
Solve a 45-45-90 triangle from a leg or the hypotenuse. Get both equal legs, the hypotenuse, area, and perimeter with instant results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
45-45-90 Triangle Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: If each leg = x, then hypotenuse = x√2
Worked example — Hypotenuse ≈ 19.799 m
Formula
If each leg = x, then hypotenuse = x√2
A 45-45-90 triangle is exactly half of a square cut along its diagonal, which permanently locks the two legs to the same length and makes the hypotenuse equal to the leg multiplied by √2. Both 45° angles are identical, so the triangle is isosceles as well as right-angled. Because the hypotenuse is the diagonal of the underlying square, this triangle appears whenever a square corner or grid layout needs to be measured diagonally.
Worked Examples
Example 1: Given a leg
Problem:A square corner forms a 45-45-90 triangle with a leg of 14 m. Find the hypotenuse.
Solution:Leg = 14 Hypotenuse = 14√2 ≈ 19.799
Result:Hypotenuse ≈ 19.799 m
Example 2: Given the hypotenuse
Problem:A diagonal segment in a right isosceles triangle is 18 in. Find each leg.
Solution:Leg = 18 ÷ √2 ≈ 12.7279 Both legs are equal in a 45-45-90 triangle.
Result:Each leg ≈ 12.7279 in
Frequently Asked Questions
What is a 45-45-90 triangle?
A 45-45-90 triangle is an isosceles right triangle. Its two legs are equal, and the hypotenuse is always the leg multiplied by √2.
What is the side ratio for a 45-45-90 triangle?
If each leg is x, then the hypotenuse is x√2. The fixed side ratio is 1 : 1 : √2.
Where is this triangle used?
45-45-90 triangles appear in squares, diagonal measurements, drafting, layout work, and many geometry and trigonometry problems.
How do I find the hypotenuse of a right triangle?
Use the Pythagorean theorem: hypotenuse = square root of (a squared + b squared), where a and b are the two legs. For a 45-45-90 triangle with legs of length 5, the hypotenuse is 5 times the square root of 2, or approximately 7.07.
What is the difference between a 45-45-90 and a 30-60-90 triangle?
A 45-45-90 triangle has two equal legs with hypotenuse = leg times sqrt(2). A 30-60-90 triangle has sides in the ratio 1 : sqrt(3) : 2, where the shortest side is opposite the 30-degree angle. Both are special right triangles with fixed side ratios.
How do I calculate the area of a triangle?
The basic formula is area = (1/2) times base times height. For a right triangle, the two legs serve as base and height. For a 45-45-90 triangle with legs of length a, the area is (1/2) times a squared.
When are special right triangles used in real life?
Special right triangles appear in construction (roof pitches, stair angles), navigation, engineering (bridge supports, truss designs), and design. A 45-degree angle creates equal rise and run, commonly used for roof slopes and diagonal bracing.
How do I know if a triangle is a right triangle?
Check if the sides satisfy the Pythagorean theorem: a squared + b squared = c squared, where c is the longest side. Common right triangle side ratios include 3:4:5, 5:12:13, 8:15:17, and 7:24:25.
How do I find a missing side of a triangle?
For right triangles, use the Pythagorean theorem. For any triangle, use the Law of Cosines (c squared = a squared + b squared - 2ab cos C) or the Law of Sines (a/sin A = b/sin B = c/sin C). Special triangles have fixed ratios that simplify calculations.
What is an isosceles right triangle?
An isosceles right triangle is the same as a 45-45-90 triangle. It has two equal legs and a right angle between them. The word isosceles means two sides of equal length. Every 45-45-90 triangle is isosceles, and every isosceles right triangle is a 45-45-90.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮30-60-90 Triangle Calculator
Solve a 30-60-90 triangle from any one side. Get the short leg, long leg, hypotenuse, area, and perimeter instantly.
🧮Triangle Inequality Theorem Calculator
Calculate triangle inequality theorem with inputs, formulas, and instant results.
🧮Pascals Triangle Calculator
Calculate pascals triangle with inputs, formulas, and instant results.
🧮Triangle Circumcenter Calculator
Calculate triangle circumcenter with inputs, formulas, and instant results.
🧮Triangle Incenter Calculator
Calculate triangle incenter with inputs, formulas, and instant results.
🧮Area of a Right Triangle Calculator
Calculate area of aright triangle with inputs, formulas, and instant results.
🧮Equilateral Triangle Calculator
Calculate equilateral triangle with inputs, formulas, and instant results.
🧮Isosceles Triangle Calculator
Calculate isosceles triangle with inputs, formulas, and instant results.