What Is an Isosceles Right Triangle?
An isosceles right triangle is a triangle that has one 90° angle and two equal legs. Because the two legs are the same length, the other two angles are both 45°, which is why it is often called a 45-45-90 triangle. It is one of the most useful special triangles in geometry, trigonometry, and construction.
How to Use This Calculator
Simply enter the length of one leg (a) — the two legs are equal, so a single value defines the whole triangle. The calculator instantly returns the hypotenuse, the area, and the perimeter. Use any consistent unit (cm, m, inches); the outputs will be in the same linear unit, and the area will be in that unit squared.
The Formulas Explained
For a leg length a:
Hypotenuse: Using the Pythagorean theorem, \(c^2 = a^2 + a^2 = 2a^2\), so $$c = a\sqrt{2}.$$
Area: The legs serve as base and height, so $$A = \tfrac{1}{2} \times a \times a = \frac{a^2}{2}.$$
Perimeter: Add all three sides: $$P = a + a + a\sqrt{2} = 2a + a\sqrt{2}.$$
Worked Example
Suppose \(a = 5\). The hypotenuse is $$5 \times \sqrt{2} \approx 7.0711.$$ The area is $$5^2 \div 2 = 12.5.$$ The perimeter is $$(2 \times 5) + 7.0711 = 17.0711.$$ So a triangle with 5-unit legs has a hypotenuse of about 7.07 units, an area of 12.5 square units, and a perimeter of about 17.07 units.
FAQ
What angles does it have? One right angle (90°) and two equal angles of 45° each.
Why is the hypotenuse a√2? Because both legs are equal, the Pythagorean theorem simplifies to \(c = \sqrt{2a^2} = a\sqrt{2}\).
Can I find the leg from the hypotenuse? Yes — divide the hypotenuse by \(\sqrt{2}\) (or multiply by \(\tfrac{\sqrt{2}}{2}\)) to get the leg length.
