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Isosceles Right Triangle Hypotenuse Calculator

Compute the hypotenuse of a 45-45-90 isosceles right triangle. Enter the leg length, select a unit, and instantly apply c = a√2 for an exact answer.

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Inputs

Leg Length (a)

Unit of Measurement

Hypotenuse Length

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Hypotenuse Length—units

The formula

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Understanding the Isosceles Right Triangle

An isosceles right triangle is a special right triangle with two equal legs and three interior angles measuring 45°, 45°, and 90°. This distinctive 45°–45°–90° configuration makes the relationship between its legs and hypotenuse entirely predictable: the hypotenuse is always exactly √2 times the length of either leg. Because of this fixed ratio, a single input — the leg length — is all that is needed to compute the hypotenuse precisely.

The Formula: c = a√2

Given a leg of length a, the hypotenuse c is calculated with:

c = a × √2 ≈ a × 1.41421356

This result follows directly from the Pythagorean theorem, which states that for any right triangle with legs a and b and hypotenuse c:

c² = a² + b²

In an isosceles right triangle, both legs are equal, so b = a. Substituting this into the theorem and simplifying yields:

c² = a² + a²
c² = 2a²
c = √(2a²)
c = a√2

The constant √2 ≈ 1.41421356… is an irrational number — its decimal expansion never terminates or repeats. Critically, it is also dimensionless, meaning the ratio holds identically whether leg lengths are measured in meters, feet, inches, centimeters, or any other unit.

Variable Reference

a — Leg Length: The length of one of the two congruent legs. Because the triangle is isosceles, both legs share this value and only one needs to be entered.
c — Hypotenuse: The side opposite the 90° angle. It is always the longest side and equals a multiplied by √2.
Unit of Measurement: The unit carries over to the result unchanged. The formula c = a√2 is purely numeric — only the numeric value of a is multiplied by √2, and the unit label is appended to the output for display purposes.

Worked Examples

Example 1: Leg = 5 cm

For a leg length of 5 cm:

c = 5 × 1.41421356 ≈ 7.071 cm

Example 2: Leg = 10 feet

For a leg length of 10 feet:

c = 10 × 1.41421356 ≈ 14.142 feet

Example 3: The Unit Square Diagonal (Leg = 1 m)

A 1 m × 1 m square cut along its diagonal produces two isosceles right triangles, each with legs of 1 m. The hypotenuse of each triangle — and the diagonal of the square — equals:

c = 1 × √2 ≈ 1.414 m

This is the classical geometric result that reveals √2 as irrational: the diagonal of a unit square cannot be expressed as a ratio of two integers. Ancient Greek mathematicians attributed this discovery to the Pythagoreans.

Example 4: Leg = 25 inches (practical construction)

A carpenter bracing a square 25-inch panel diagonally needs a brace of:

c = 25 × 1.41421356 ≈ 35.355 inches

Practical Applications

The 45°–45°–90° triangle appears widely in architecture, engineering, design, and everyday geometry:

Construction and Carpentry: Calculating diagonal brace lengths for square frames, panels, and shelving units.
Flooring and Tiling: Determining cut lengths when square tiles are installed at a 45° angle to room walls.
Roof Framing: Symmetrical hip roofs form 45°–45°–90° triangles at corner intersections, and the rafter lengths follow c = a√2.
Surveying and Navigation: Estimating straight-line distances across square lots, city blocks, or land parcels laid out on a grid.
Computer Graphics and Game Design: Diagonal movement in grid-based systems travels √2 times the cell size per step, requiring this formula for accurate speed normalization.
Paper Sizes: The ISO 216 standard (A4, A3, A2, etc.) uses an aspect ratio of 1 : √2, which allows each sheet to be folded exactly in half to produce the next smaller size.

Sources and Methodology

The derivation above applies the Pythagorean theorem to the special case of equal legs, as documented in Wolfram MathWorld: Isosceles Right Triangle, the leading peer-reviewed online mathematical reference. A full pedagogical treatment of 45°–45°–90° triangles within right triangle trigonometry appears in Chapter 8: Trigonometry of the Right Triangle (Herricks UFSD Algebra Textbook). Additional algebraic derivations involving isosceles right triangles in quadratic equation contexts are provided in CUNY Borough of Manhattan Community College Mathematics Department course notes on quadratic equations.

Reference

Frequently asked questions

What is an isosceles right triangle?

An isosceles right triangle is a triangle with exactly one 90° angle and two equal-length legs. Because interior angles must sum to 180°, the remaining two angles are each 45°, producing the characteristic 45°–45°–90° pattern. The two equal legs can be any positive length, and the hypotenuse is always √2 (approximately 1.41421356) times the leg length. It is also the exact shape formed by cutting a square in half along either diagonal.

How do you calculate the hypotenuse of an isosceles right triangle?

Multiply the leg length by √2 ≈ 1.41421356. For example, a leg of 7 meters gives a hypotenuse of 7 × 1.41421356 ≈ 9.899 meters; a leg of 12 inches gives 12 × 1.41421356 ≈ 16.971 inches. The formula c = a√2 is derived from the Pythagorean theorem: since both legs equal a, c² = a² + a² = 2a², so c = a√2. This calculation is exact and applies at any scale.

Why is the hypotenuse of a 45-45-90 triangle always √2 times the leg?

The Pythagorean theorem states c² = a² + b². In a 45°–45°–90° triangle both legs are equal (a = b), so c² = a² + a² = 2a². Taking the positive square root of both sides gives c = √2 × a. This multiplier is fixed entirely by the equal-leg geometry of the triangle. Whether the leg is 1 millimeter or 1 kilometer, the ratio between the hypotenuse and each leg is always the irrational constant √2 ≈ 1.41421356.

Does the unit of measurement affect the hypotenuse calculation?

No — the unit does not change the formula. The ratio √2 is a pure dimensionless number. A leg of 5 meters yields a hypotenuse of 5√2 meters (≈ 7.071 m); a leg of 5 inches yields 5√2 inches (≈ 7.071 in). The numeric result scales with the magnitude of the input, but the multiplier √2 ≈ 1.41421356 is identical in every system of units. Simply keep the unit consistent across input and output.

What are the three angles in an isosceles right triangle?

The three interior angles are 90°, 45°, and 45°. The right angle (90°) sits at the vertex where the two equal legs meet and is opposite the hypotenuse. Each of the two base angles measures 45° because the isosceles property forces them to be equal, and they must together account for the remaining 90°. Together, 90° + 45° + 45° = 180°, satisfying the triangle angle-sum theorem for all Euclidean triangles.

How does an isosceles right triangle differ from a 30-60-90 triangle?

Both are special right triangles with fixed side ratios, but their proportions differ significantly. A 45°–45°–90° triangle has sides in the ratio 1 : 1 : √2 and is the only right triangle that is also isosceles (two sides equal). A 30°–60°–90° triangle has sides in the ratio 1 : √3 : 2 and is scalene (all three sides different). For a shared leg of length a, the 45-45-90 hypotenuse is a√2 ≈ 1.414a, while the 30-60-90 hypotenuse is 2a — noticeably longer.