Right Triangle Hypotenuse Calculator

Formula

c = √(a² + b²)

Worked example

With First leg: 3 units, Second leg: 4 units, hypotenuse is 5 units.

Calculation method

The rule is c = √(a² + b²).

The Pythagorean theorem squares the two perpendicular legs, adds them, then takes the square root. The hypotenuse is always the longest side.

For this right triangle hypotenuse example, changing first leg from 3 units to 4 units while keeping the other inputs fixed changes hypotenuse from 5 units to 5.656854 units.

Reading hypotenuse

A triangle’s area is half the corresponding base-times-height rectangle. In a right triangle, the hypotenuse is opposite the right angle and is the longest side. In the worked values on this page, first leg 3 units, second leg 4 units produce hypotenuse 5 units.

Practical use

Use these formulas for surveying, drawings, roof geometry, and checking right-angle layouts. The specific task here is to find hypotenuse from first leg, second leg.

Limits and common mistakes

The height must be perpendicular to the selected base. Three side lengths must satisfy the triangle inequality to form a real triangle. Verify first leg in units before interpreting hypotenuse; a value measured on another basis would answer a different question.

Why the relationship works

The Pythagorean relationship applies only when the two legs meet at a right angle. Adding their squares gives the square of the hypotenuse; subtracting a known leg’s square from the hypotenuse squared recovers the other leg. This operation involves lengths, so take the appropriate square root at the end. The right triangle hypotenuse rule uses first leg (units), second leg (units) in c = √(a² + b²); read each symbol as the field named on this page.

Interpreting another set of values

For right triangle hypotenuse, the fields labeled first leg (units), second leg (units) are the known quantities. The calculator solves for hypotenuse in units using c = √(a² + b²). Label the actual base, perpendicular height, side lengths, or right-angle legs before using the answer. A result for area cannot be compared directly with a side length because their units differ. When comparing two calculations, keep their units and input definitions identical.

For a second right triangle hypotenuse check, keep the other default inputs fixed and change first leg from 3 units to 5 units. Hypotenuse changes from 5 units to 6.403124 units. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

The legs must meet at a right angle, and the hypotenuse must be longer than either leg. When finding a missing leg, a known leg longer than the hypotenuse cannot describe a right triangle. All lengths should use the same unit. The reported hypotenuse is 5 units in the worked example. Keep its stated unit or counting convention when checking it against another calculation.