Free Isosceles Right Triangle Hypotenuse Calculator

The Isosceles Right Triangle Hypotenuse Calculator is a free online geometry tool designed to compute the hypotenuse length from the given leg measurement. It acts as a focused Pythagorean theorem calculator for 45-45-90 triangles, delivering results using the core triangle hypotenuse formula. This practical right triangle hypotenuse solver serves students, engineers, and hobbyists who need quick and accurate answers.

What Is an Isosceles Right Triangle?

An isosceles right triangle has two equal sides (the legs) and one interior angle of exactly 90°. Because it satisfies the definitions of both an isosceles triangle and a right triangle, it occupies a unique place among geometric figures. The sum of interior angles is always 180°, so with one 90° angle, the other two angles are equal and each equals 45°. This special triangle is often called a 45-45-90 triangle. Its side lengths follow a fixed ratio: the legs are in proportion $1 : 1$, and the hypotenuse is $\sqrt{2}$ times the leg length.

Hypotenuse Formula Derived from the Pythagorean Theorem

The Pythagorean theorem is the standard tool for right triangle calculations. For a triangle with legs $B$ and hypotenuse $A$, the theorem states:

$$A^{2} = B^{2} + C^{2}$$

In an isosceles right triangle, the two legs are equal, so $C = B$. Substituting gives:

$$A^{2} = B^{2} + B^{2} = 2B^{2}$$

Solving for $A$ yields the simplified formula:

$$A = B \times \sqrt{2}$$

This relationship is the foundation of every isosceles right triangle hypotenuse calculation.

Step-by-Step Calculation Example

Suppose the leg length is 5 cm. The hypotenuse is found as:

1. Square the leg: $5^{2} = 25$.
2. Double the result: $2 \times 25 = 50$.
3. Take the square root: $\sqrt{50} \approx 7.071$.

Thus the hypotenuse is approximately 7.07 cm. The same three steps apply to any leg measurement.

Quick-Reference Table for Common Leg Lengths

The table below shows the hypotenuse for several typical leg lengths, computed with the formula.

Leg ($B$)	Hypotenuse ($A \approx$)
1	1.414
2	2.828
3	4.243
5	7.071
10	14.142

This pattern illustrates how the hypotenuse scales proportionally with the leg.

How the Calculator Works

Using this geometry calculator is simple:

• Enter the leg length (B) into the input field.
• The tool automatically applies the relation $A = B\sqrt{2}$ and outputs the hypotenuse.

The calculator eliminates the need for manual square roots and reduces calculation errors. If you need to work backward from the hypotenuse to the leg length, the same relationship can be rearranged to $B = A / \sqrt{2}$, offering a complete solution for both directions.

Why Use This Tool?

Manual application of the Pythagorean theorem is time-consuming and error-prone when square roots are involved. This calculator provides instant, reliable results, making it ideal for homework, design projects, and any scenario requiring a fast and trustworthy right triangle hypotenuse calculation.