Isosceles Right Triangle Calculator

Enter either the leg length or the hypotenuse of a 45-45-90 isosceles right triangle to get the leg, hypotenuse, area (A = a²/2), and perimeter.

Quick Facts

Leg-hypotenuse relation
c = a√2 ≈ 1.4142a
Both legs are equal length; the hypotenuse follows from the Pythagorean theorem, a² + a² = c².
Area formula
A = a² / 2
The two equal legs meet at the right angle, so one is the base and the other the height.
Angles
45° - 45° - 90°
Equal legs mean the two acute angles are always equal, at 45° each.

Your Results

Calculated
Leg Length (a)
-
Each of the two equal legs
Hypotenuse (c)
-
c = a√2
Area
-
A = a² / 2
Perimeter
-
P = 2a + c

Ready

Choose which dimension you know, enter its value and unit, then press Calculate.

Formula and Method for the Isosceles Right Triangle

An isosceles right triangle has one 90° angle and two legs of equal length, which forces the two remaining angles to also be equal — 45° each. It is commonly called a 45-45-90 triangle. Because two sides and the included angle are always known relationships, every other measurement (the hypotenuse, area, and perimeter) can be derived from a single known dimension: either a leg or the hypotenuse.

How the calculation works

Enter either the leg length (a) or the hypotenuse (c) and choose the unit. If you enter a leg, the calculator finds the hypotenuse using the Pythagorean theorem: since both legs equal a, a² + a² = c², so c = a√2 ≈ 1.4142 × a. If you enter the hypotenuse instead, it solves for the leg with a = c / √2 = c√2 / 2 ≈ 0.7071 × c. Once both legs and the hypotenuse are known, the area follows from treating the two equal legs as the base and height (they meet at the right angle): A = a² / 2. The perimeter is simply the sum of all three sides: P = 2a + c.

Common mistakes

  • Assuming the hypotenuse when a leg is given, or vice versa: double-check which side you actually measured before entering it — the leg and hypotenuse formulas are reciprocals of each other (multiply by √2 vs. divide by √2).
  • Using the general triangle area formula incorrectly: the shortcut A = a²/2 only works because the two legs are perpendicular. For a general triangle you would need A = ½ × base × height, with height measured perpendicular to the chosen base.
  • Mixing units: keep the length in one consistent unit — convert inches to feet, or centimeters to meters, before entering the value; area results come out in that same unit squared.

Real-world applications

  • Carpentry and framing use 45-45-90 triangles constantly for mitre joints, corner braces, and diagonal bracing.
  • CNC and laser cutting layouts often specify a leg length and rely on the √2 relationship to cut the hypotenuse edge precisely.
  • Physics and engineering use the 45° isosceles right triangle to resolve a vector into equal perpendicular components.
  • Architecture and design use it for stair stringers, roof pitches, and gusset plates where a 45° corner is required.

Frequently Asked Questions

What is an isosceles right triangle?
An isosceles right triangle has one 90° angle and two legs of equal length. Because the legs are equal, the two remaining angles are also equal — each measuring 45°. It is often called a 45-45-90 triangle.
How do I find the hypotenuse from the leg length?
Multiply the leg length by the square root of 2: c = a√2 ≈ 1.4142 × a. This follows directly from the Pythagorean theorem, a² + a² = c², since both legs are the same length a.
How do I find the leg length from the hypotenuse?
Divide the hypotenuse by the square root of 2: a = c / √2, which is the same as a = c√2 / 2 ≈ 0.7071 × c.
What is the area formula for an isosceles right triangle?
Area equals half the product of the two equal legs: A = a² / 2. This works because the two legs meet at a 90° angle, so one leg is the base and the other is the height.

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