Calculator Hypotenuse Length: Formula & Steps

This calculator finds the hypotenuse length of a right triangle when you enter the lengths of the two legs (the sides that form the right angle). It uses the Pythagorean theorem to return the hypotenuse in the same unit you entered, along with the perimeter and area as helpful extras.

Formula

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

c = √(a² + b²)

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

The formula works for any unit as long as both legs are measured in the same unit. The result is in that same unit.

Worked Example

Using the default values from the calculator: leg a = 3, leg b = 4.

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

The hypotenuse is 5. This is the classic 3-4-5 triangle, often used in construction to check right angles. The perimeter is 3 + 4 + 5 = 12, and the area is (3 × 4) / 2 = 6.

When to Use the Hypotenuse Calculator

You need the hypotenuse whenever you know the two legs of a right triangle and want the longest side. Common situations include:

• Finding the diagonal of a rectangle or square (the diagonal splits the rectangle into two right triangles)
• Determining the length of a ladder needed to reach a certain height against a wall
• Calculating the distance between two points on a flat map when you know the north-south and east-west distances
• Checking if a corner is square (if the sides satisfy a² + b² = c², the angle is 90°)
• Finding the length of a ramp or roof slope when you know the horizontal run and vertical rise

If you only know one leg and the hypotenuse, you can rearrange the formula to find the missing leg: a = √(c² – b²).

How to Read the Result

The hypotenuse is always the longest side of a right triangle. It will always be longer than either leg, but shorter than the sum of the two legs. For example, with legs 3 and 4, the hypotenuse is 5, which is between 4 and 7.

The calculator also shows the perimeter (the total distance around the triangle) and the area (the space enclosed). These are useful if you are planning to build something or need to cover the triangle with material.

Common Mistakes and Misconceptions

• Mixing units: If one leg is in feet and the other in inches, the result is wrong. Convert both to the same unit first.
• Forgetting to square both legs: Some people add the legs and then square the sum. That is incorrect. You must square each leg separately, then add.
• Applying the theorem to non-right triangles: The Pythagorean theorem only works for right triangles (one angle exactly 90°). For other triangles, use the law of cosines.
• Rounding too early: If you round intermediate values, the final result may be off. Keep full precision until the final step.

A common misconception is that the hypotenuse is always the side opposite the 90° angle, which is true, but some people think it is the longest side of any triangle. In an obtuse triangle, the longest side is opposite the obtuse angle, not a hypotenuse (since no right angle exists).

Limitations of the Formula

The Pythagorean theorem assumes a perfectly flat, Euclidean plane. It does not account for Earth’s curvature, which matters for very large distances (hundreds of kilometers). It also assumes the triangle is exactly right-angled; if the angle is slightly off, the result is an approximation.

For real-world measurements, the accuracy depends on the precision of your leg measurements. If you measure with a tape that has 1 cm error, the hypotenuse will have a similar magnitude of error.

Frequently Asked Questions

How do I find the hypotenuse without a calculator?

Square both legs, add them, and take the square root of the sum. For example, legs 6 and 8: 36 + 64 = 100, √100 = 10. You can do this by hand or with a basic calculator.

What if I only know one leg and the hypotenuse?

Rearrange the formula to find the missing leg: a = √(c² – b²). For example, if hypotenuse c = 13 and leg b = 5, then a = √(169 – 25) = √144 = 12.

Does the Pythagorean theorem work for all triangles?

No, only for right triangles. For any triangle, you can use the law of cosines: c² = a² + b² – 2ab·cos(C), where C is the angle between sides a and b. When C = 90°, cos(C) = 0, so the formula simplifies to the Pythagorean theorem.

What is the hypotenuse of a 45-45-90 triangle?

In an isosceles right triangle (45-45-90), the legs are equal. If each leg is length L, the hypotenuse is L√2. For example, legs of 1 give a hypotenuse of √2 ≈ 1.414.

Can the hypotenuse be negative?

No. Length is always positive. The calculator enforces non-negative inputs, and the square root of a sum of squares is always non-negative.

How do I measure the hypotenuse in real life?

If you can measure the two legs directly, use the formula. If you cannot measure the legs, you can use a tape measure along the diagonal, but that only works if the triangle is physically accessible. For large structures, laser distance meters or surveying equipment can measure the hypotenuse directly.

Related

• Area Calculator for Rectangle: Formula & Examples
• Area Calculator Hectares: How to Convert & Measure
• Cube Area Calculator: Find Surface Area & Volume

in a | Leg a | - | 3 | 0
in b | Leg b | - | 4 | 0
out c | Hypotenuse | - | sqrt(a^2 + b^2) | 2
out per | Perimeter | - | a + b + c | 2
out area | Area | - | (a * b) / 2 | 2