Isosceles Right Triangle Hypotenuse Calculator

Enter one known side of a 45-45-90 (isosceles right) triangle — a leg or the hypotenuse — to get the hypotenuse, leg length, area, and perimeter using h = a√2.

Quick Facts

Hypotenuse formula
h = a√2
From the Pythagorean theorem a² + a² = h², since both legs equal a.
Leg from hypotenuse
a = h / √2
Divide the hypotenuse by √2 ≈ 1.4142 to get each leg.
Angles
45° - 45° - 90°
Both legs are equal, so the two acute angles are always 45°.
Area formula
A = a² / 2
Half the product of the two equal legs (base × height ÷ 2).

Your Results

Calculated
Hypotenuse (h)
-
h = a√2
Leg Length (a)
-
a = h / √2
Area
-
A = a² / 2
Perimeter
-
P = 2a + h

Ready

Choose the known side, enter its value and unit, then press Calculate.

Formula and Method for the Isosceles Right Triangle Hypotenuse

An isosceles right triangle — also called a 45-45-90 triangle — has two legs of equal length meeting at a right angle. Because the legs are equal, the Pythagorean theorem simplifies to a single relationship between a leg and the hypotenuse: h = a√2, where a is the length of each leg and h is the hypotenuse (the side opposite the right angle). This calculator works in either direction: enter a leg to find the hypotenuse, or enter the hypotenuse to find the leg, along with the triangle's area and perimeter.

How the calculation works

Starting from the Pythagorean theorem a² + b² = c², an isosceles right triangle has a = b, so a² + a² = h², or 2a² = h². Taking the square root of both sides gives h = a√2 ≈ 1.4142 × a. To go the other direction, solve for a: a = h / √2, which is the same as a = h√2 / 2 after rationalizing the denominator. Once both the leg and hypotenuse are known, the area is A = a² / 2 (half the product of the two legs, which form the base and height), and the perimeter is P = 2a + h.

Common mistakes

  • Applying √2 to the wrong side: multiply the leg by √2 to get the hypotenuse; divide the hypotenuse by √2 to get the leg. Doing it backward gives a leg longer than the hypotenuse, which is impossible.
  • Confusing this with a general isosceles triangle: the h = a√2 relationship only applies when the two equal sides meet at a right angle. A general isosceles triangle with a different apex angle needs the Law of Cosines instead.
  • Mixing units: keep the leg length and hypotenuse in the same unit system throughout; convert before entering a value.

Real-world applications

  • Carpentry and framing use 45-45-90 triangles for mitered corners, braces, and diagonal supports where two equal legs meet at a right angle.
  • Cutting a square diagonally in half produces two isosceles right triangles, useful for gussets, roof trusses, and gable ends.
  • Surveying and navigation use the relationship to find a diagonal distance when two perpendicular legs (such as north-south and east-west offsets) are equal.
  • Graphic design and CAD layouts rely on the 45° diagonal to build symmetric shapes and true square corners.

Frequently Asked Questions

What is the formula for the hypotenuse of an isosceles right triangle?
In a 45-45-90 (isosceles right) triangle, both legs are the same length a, so the Pythagorean theorem a² + a² = h² simplifies to h = a√2 ≈ 1.4142 × a. A triangle with 5 ft legs has a hypotenuse of 5 × √2 ≈ 7.07 ft.
How do I find the leg length if I know the hypotenuse?
Divide the hypotenuse by √2: a = h / √2, which is the same as a = h√2 / 2. A hypotenuse of 10 in gives legs of 10 / √2 ≈ 7.07 in each.
How do you calculate the area of an isosceles right triangle?
Area equals half the product of the two legs, and since both legs equal a, A = a² / 2. For legs of 6 cm, the area is 6² / 2 = 18 cm².
Why are the two acute angles always 45°?
Because the two legs are equal, the base angles opposite them are equal too. The three angles sum to 180°, and one angle is 90°, so the remaining 90° splits evenly into two 45° angles.

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