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Hypotenuse Calculator

Calculate the hypotenuse of a right triangle from two legs. Free step-by-step solution with interactive diagram and Pythagorean triple detection.

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Try an Example

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Ladder Against Wall

A 10-foot ladder leans against a wall with the base 6 feet away. How high does it reach?

Key values: Leg a = 6 ft · Hypotenuse c = 10 ft · Solve for leg b

TV Screen Diagonal

Find the diagonal of a TV with a 16:9 aspect ratio (width 40", height 22.5").

Key values: Width = 40 in · Height = 22.5 in · Find diagonal

Classic 3-4-5 Triangle

Verify that sides 3, 4, and 5 form a right triangle (the most famous Pythagorean triple).

Key values: Sides: 3, 4, 5 · Right triangle · Primitive triple

Documentation

Finding the Hypotenuse

The hypotenuse is the longest side of a right triangle — the side opposite the right angle. Given legs $a$ and $b$ :

c = \sqrt{a^2 + b^2}

This is a direct application of the Pythagorean theorem, solving for $c$ .

Special Right Triangles

Triangle type	Sides ratio	Example	Angles
45-45-90	$1 : 1 : \sqrt{2}$	5, 5, 7.07	45°, 45°, 90°
30-60-90	$1 : \sqrt{3} : 2$	5, 8.66, 10	30°, 60°, 90°
3-4-5	3 : 4 : 5	6, 8, 10	≈ 37°, 53°, 90°
5-12-13	5 : 12 : 13	10, 24, 26	≈ 23°, 67°, 90°

Quick estimate: For a 45-45-90 triangle, the hypotenuse is about 1.414× the leg length. For a 30-60-90, it's exactly 2× the shortest side.

Finding a Leg (Not the Hypotenuse)

If you know the hypotenuse $c$ and one leg $a$ , solve for the other leg:

b = \sqrt{c^2 - a^2}

Common mistake: Forgetting that this uses subtraction, not addition. The hypotenuse is always the largest side, so $c^2 - a^2$ is always positive.

Real-World Applications

Screen diagonals: A 16:9 monitor with 24" width and 13.5" height has a diagonal of $\sqrt{24^2 + 13.5^2} \approx 27.5$ "
Ladder safety: A 20-foot ladder reaching 16 feet up a wall should be $\sqrt{20^2 - 16^2} = 12$ feet from the base
Navigation: Walking 3 blocks east and 4 blocks north puts you $\sqrt{9 + 16} = 5$ blocks from start (as the crow flies)

Frequently Asked Questions

How do I find the hypotenuse of a right triangle?

Use the formula $c = \sqrt{a^2 + b^2}$ , where $a$ and $b$ are the two legs. Square each leg, add the results, and take the square root. For example, legs of 3 and 4 give $c = \sqrt{9 + 16} = \sqrt{25} = 5$ .

What are the most common Pythagorean triples?

The most common are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also valid: $(6, 8, 10) = 2 \times (3, 4, 5)$ . Memorizing these speeds up calculations when exact integer answers exist.

What are 45-45-90 and 30-60-90 triangles?

These are special right triangles with fixed side ratios. A 45-45-90 triangle has sides in the ratio $1 : 1 : \sqrt{2}$ , so the hypotenuse is about 1.414 times a leg. A 30-60-90 triangle has sides in the ratio $1 : \sqrt{3} : 2$ , so the hypotenuse is exactly twice the shortest side.

How do I find a missing leg instead of the hypotenuse?

Rearrange the formula: $b = \sqrt{c^2 - a^2}$ . Subtract the square of the known leg from the square of the hypotenuse, then take the square root. Remember to subtract (not add) because you are solving for a shorter side.

Can the Pythagorean theorem be used with non-right triangles?

No. The formula $a^2 + b^2 = c^2$ applies only to right triangles. For non-right triangles, use the Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$ , which reduces to the Pythagorean theorem when $C = 90^\circ$ .

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