Isosceles Right Triangle
Quickly solve any 45-45-90 isosceles right triangle from leg length. Get hypotenuse, area, and perimeter instantly with this free online geometry calculator.
About This Calculator
This free online isosceles right triangle calculator solves a 45-45-90 triangle from a single leg length. In an isosceles right triangle, the two legs are equal and the hypotenuse equals leg \u00d7 \u221a2. The calculator also computes area (\u00bd \u00d7 leg\u00b2) and perimeter (2 \u00d7 leg + hyp). Simply enter the leg length to instantly get all triangle properties.
Isosceles right triangles appear frequently in geometry, trigonometry, and design. They are common in standardized tests (SAT, ACT, GRE) and real-world applications like roof trusses, square corner measurements, and 45\u00b0 miter cuts in woodworking. The 45-45-90 triangle is one of the two special right triangles (along with the 30-60-90 triangle) taught in high school geometry.
The calculator uses the following formulas: Hypotenuse = leg \u00d7 \u221a2, Area = \u00bd \u00d7 leg\u00b2, and Perimeter = 2 \u00d7 leg + hypotenuse = leg(2 + \u221a2). These formulas derive directly from the Pythagorean theorem (hypotenuse\u00b2 = leg\u00b2 + leg\u00b2 = 2 \u00d7 leg\u00b2) and the properties of an isosceles triangle where the two base angles are both 45\u00b0. Since the legs are equal, the area formula simplifies to half the square of a leg.
The hypotenuse is always \u221a2 (approximately 1.414) times the leg length. This ratio is constant for all 45-45-90 triangles, making them similar to each other. For example, if the leg is 10 units, the hypotenuse is about 14.14 units, the area is 50 square units, and the perimeter is about 34.14 units. The special constant \u221a2 is an irrational number approximately equal to 1.41421356237, and this calculator provides results rounded to two decimal places for convenience.
Frequently Asked Questions
What is an isosceles right triangle?
An isosceles right triangle (45-45-90) has two equal legs and a right angle. The two acute angles are both 45\u00b0. The hypotenuse equals leg \u00d7 \u221a2.
How do you find the hypotenuse from the leg?
Multiply the leg length by \u221a2 (approximately 1.414). For example, if the leg is 5, the hypotenuse is 5 \u00d7 1.414 = 7.07.
How do you calculate the area?
Area = \u00bd \u00d7 leg\u00b2, since both legs are equal. For a leg of 5, the area is \u00bd \u00d7 25 = 12.5 square units.
How do you calculate the perimeter?
Perimeter = 2 \u00d7 leg + hypotenuse = 2 \u00d7 leg + leg \u00d7 \u221a2 = leg(2 + \u221a2).
Is this tool free?
Yes, all calculators on Calculy are completely free to use with no registration required.
What is the ratio of the sides in a 45-45-90 triangle?
The side ratio in a 45-45-90 triangle is 1 : 1 : \u221a2. This means if the legs are 1 unit each, the hypotenuse is \u221a2 \u2248 1.414 units. This constant ratio makes all isosceles right triangles similar to each other.
What is the Pythagorean theorem for isosceles right triangles?
For an isosceles right triangle with leg length a, the Pythagorean theorem simplifies to hyp\u00b2 = a\u00b2 + a\u00b2 = 2a\u00b2, so the hypotenuse is a\u00d7\u221a2. This is the same formula used in this calculator.
How do you find the missing leg if you know the hypotenuse?
If you know the hypotenuse h, the leg length is h / \u221a2. For example, if the hypotenuse is 10, each leg is 10 / 1.414 \u2248 7.07 units. Use our companion calculator Isosceles Right Triangle Hypotenuse Calculator for this case.
What is the circumradius of an isosceles right triangle?
The circumradius R of an isosceles right triangle is half the hypotenuse: R = hyp / 2 = (leg \u00d7 \u221a2) / 2 = leg / \u221a2. Since the circumcenter of a right triangle lies at the midpoint of the hypotenuse.
What is the inradius of an isosceles right triangle?
The inradius r of an isosceles right triangle equals (2 - \u221a2) \u00d7 leg / 2 \u2248 0.2929 \u00d7 leg. This is derived from the formula r = 2A / P where A is area and P is perimeter.