Hypotenuse Calculator: Find Length Fast

This calculator finds the length of the hypotenuse (the longest side) of a right triangle when you enter the lengths of the other two sides. It uses the Pythagorean theorem, a² + b² = c², and returns the hypotenuse in the same unit you entered. You also get the triangle’s area and perimeter as a bonus.

Formula

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b):

c² = a² + b²

To solve for the hypotenuse, take the square root of both sides:

c = √(a² + b²)

• c = hypotenuse (the side opposite the right angle), in the same unit as a and b
• a = one leg (short side) of the triangle, in any unit
• b = the other leg, in the same unit as a

The area is (a × b) / 2, and the perimeter is a + b + c.

Worked Example

Using the default values: a = 3 m, b = 4 m.

1. Square each leg: 3² = 9, 4² = 16.
2. Add the squares: 9 + 16 = 25.
3. Take the square root: √25 = 5.

The hypotenuse is 5 m (rounded to 2 decimals, it shows 5.00). The area is 0.5 × 3 × 4 = 6 m², and the perimeter is 3 + 4 + 5 = 12 m.

When to Use This Calculator

Use it whenever you have a right triangle and know two sides. Common situations include:

• Finding the length of a ladder needed to reach a certain height on a wall (ladder = hypotenuse, wall height and floor distance are the legs).
• Computing the diagonal of a rectangle or TV screen (length and width are the legs).
• Determining the shortest distance between two points on a flat map when moving along perpendicular directions.
• Checking if a corner is square: if a triangle with sides 3, 4, and 5 fits, the corner is 90°.

The formula only works for right triangles (one angle exactly 90°). If the triangle isn’t right, you need the law of cosines instead.

How to Read the Result

The hypotenuse is always the longest side, so it should be larger than either leg. If you enter both legs in meters, the hypotenuse is in meters; if you enter feet, it’s in feet. The area is in square units (m², ft², etc.) and the perimeter is in the same linear unit.

For example, a right triangle with legs 6 cm and 8 cm gives a hypotenuse of 10 cm. That’s a 3-4-5 ratio scaled by 2. Many right triangles have irrational hypotenuses, like legs 1 and 1 giving √2 ≈ 1.414. The calculator rounds to 2 decimals, but you can change that in the output settings if you need more precision.

Common Mistakes and Unit Mix-ups

• Mixing units: If one leg is in meters and the other in centimeters, the result is wrong. Convert both to the same unit first (e.g., 1 m = 100 cm).
• Using the wrong side: The hypotenuse is always opposite the 90° angle, not just the longest side you see. If you don’t have a right angle, this formula doesn’t apply.
• Forgetting to square the numbers: You can’t just add a and b and take the square root. For 3 and 4, 3+4=7, and √7 ≈ 2.65, which is wrong. Always square first.
• Rounding too early: If you round intermediate values, the final answer may be off. Keep full precision until the last step.

Limits and Extensions

This calculator only handles right triangles. If you have an oblique triangle (no 90° angle), you need the law of cosines: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. For non-right triangles, you also need at least one angle, not just two sides.

The Pythagorean theorem is a special case of the law of cosines when C = 90°, because cos(90°) = 0. The theorem has been known for thousands of years and is one of the most fundamental relationships in geometry.

In real-world measurement, the triangle might not be perfectly right due to measurement error. The result is only as accurate as your inputs. If you measure a wall and floor at 3.00 m and 4.00 m, the hypotenuse is 5.00 m, but if your measurements are off by 1 cm, the hypotenuse could be off by about 0.7 cm.

Frequently Asked Questions

How do I find the hypotenuse if I only know one leg and one angle?

If you know one leg and an acute angle (not the right angle), use trigonometry. If the known leg is adjacent to the angle, hypotenuse = adjacent / cos(angle). If it’s opposite, hypotenuse = opposite / sin(angle). Make sure your calculator is in the correct mode (degrees or radians).

What if the triangle is not a right triangle?

Then the Pythagorean theorem doesn’t apply. You need the law of cosines, which requires two sides and the included angle, or other combinations with the law of sines. This calculator is only for right triangles.

Can I use this for finding the diagonal of a rectangle?

Yes. The diagonal of a rectangle forms a right triangle with the length and width as legs. For a rectangle 10 m by 24 m, the diagonal is √(10² + 24²) = √676 = 26 m.

Why is the hypotenuse always the longest side?

Because it’s opposite the largest angle (the 90° angle). In any triangle, the longest side is opposite the largest angle, and the right angle is the largest angle in a right triangle.

What is the 3-4-5 rule?

It’s a quick way to check for right angles using whole numbers. If a triangle has sides 3, 4, and 5 (in any unit), it’s a right triangle because 3² + 4² = 5². Multiples like 6-8-10 or 9-12-15 also work. Builders and carpenters often use this to verify square corners.

Can the hypotenuse be negative?

No. Length is always positive. The calculator enforces non-negative inputs, and the square root of a positive number is positive. If you enter a negative leg length, the result would be invalid, so the calculator prevents it.

Related

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in a | Side a | m | 3 | 0
in b | Side b | m | 4 | 0
out c | Hypotenuse | m | sqrt(a^2 + b^2) | 2
out area | Area | m² | 0.5 * a * b | 2
out perim | Perimeter | m | a + b + sqrt(a^2 + b^2) | 2