Hypotenuse Calculation Steps and Formula

To calculate a right triangle’s hypotenuse, square the lengths of its two legs, add those squares, and take the square root: c = √(a² + b²). Enter the two leg lengths to find the hypotenuse, along with the triangle’s area as a useful secondary result; the legs must meet at a right angle.

Formula

The Pythagorean theorem gives the hypotenuse of a right triangle:

c = √(a² + b²)

• a is the length of one leg, in any length unit.
• b is the length of the other leg, in the same unit as a.
• c is the hypotenuse, the side opposite the right angle, in that same unit.

The calculator’s additional area result uses Area = ab / 2. Its units are square units: for example, square metres if the legs are in metres, or square feet if they are in feet. The hypotenuse formula itself does not depend on a particular measurement system.

Worked Example

Using the calculator’s default values, a = 3 units and b = 4 units:

1. Square each leg: 3² = 9 and 4² = 16.
2. Add the squares: 9 + 16 = 25.
3. Take the square root: c = √25 = 5 units.
4. For the secondary result, area = (3 × 4) / 2 = 6 square units.

The hypotenuse is 5.00 units, and the area is 6.00 square units, matching the calculator’s two-decimal display. These lengths form the familiar 3–4–5 right triangle.

Step-by-Step Calculation

For any right triangle, follow the same sequence. The names of the legs do not matter: either one can be a or b.

1. Identify the right angle. Confirm that the two sides you are using as inputs meet at 90 degrees. Those sides are the legs.
2. Measure both legs. Use the same unit for each measurement, such as centimetres and centimetres or inches and inches.
3. Square each length. Multiply each leg by itself.
4. Add the squared values. This gives the square of the hypotenuse.
5. Take the positive square root. A side length is positive, so use the positive result.
6. Round only at the end. Keep extra digits during the calculation if you need a precise answer.

For example, if the legs are 6 cm and 8 cm, then c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm. The result is in centimetres because both inputs were in centimetres.

Why the Theorem Works

The Pythagorean theorem describes a relationship among the three sides of a right triangle. The squares built on the two legs have areas a² and b², and their combined area equals the area of the square built on the hypotenuse, c². Rearranging c² = a² + b² and taking the square root gives the calculation for c.

This relationship applies specifically to right triangles in ordinary Euclidean geometry. It does not say that the longest side of every triangle can be calculated by adding the squares of the other two sides. The 90-degree angle is essential.

When to Use It and How to Check the Result

Use the theorem when you know both legs of a right triangle and need the diagonal side. It can apply to a rectangular frame’s diagonal, a right-angled ramp cross-section, or the straight-line diagonal across a rectangular surface. In each case, the two perpendicular dimensions are the legs, and the diagonal is the hypotenuse.

A quick check is that the hypotenuse must be longer than either leg. For positive legs, √(a² + b²) is greater than both a and b. If your answer is shorter than a leg, check the arithmetic, the side labels, and whether you accidentally divided or subtracted instead of adding.

For a rectangle with width w and height h, its diagonal is √(w² + h²). This is the same theorem: the width and height form the perpendicular legs. For a three-dimensional box, one application finds a face diagonal; finding the space diagonal requires another step or an extended formula, √(l² + w² + h²), where all three dimensions are perpendicular.

Common Mistakes and Limits

• Using sides that do not meet at 90 degrees: The theorem cannot directly find a side in an arbitrary triangle. A different method, such as the law of cosines, may be needed when the included angle is known.
• Mixing units: Convert measurements before calculating. For example, change 1 metre and 30 centimetres to a common unit rather than combining their numerical values as if both were centimetres.
• Forgetting to square both legs: The expression is a² + b², not a + b or (a + b)².
• Taking the square root too early: Add the squared values first, then take one square root.
• Rounding intermediate values: Early rounding can slightly alter the final result, especially when measurements are not whole numbers.
• Treating measured inputs as exact: A calculator can provide many decimal places, but those digits do not make approximate measurements more precise. Report a result that suits the accuracy of the measurements.

There is also a useful reverse check: if all three sides are known, a triangle is right-angled when the square of its longest side equals the sum of the squares of the other two. For instance, 5² = 3² + 4², or 25 = 9 + 16.

Frequently Asked Questions

What is the formula for finding a hypotenuse?
For legs a and b of a right triangle, use c = √(a² + b²). The hypotenuse c lies opposite the 90-degree angle.

Can I find the hypotenuse with only one leg?
Not in general. You also need the other leg, or other sufficient information such as an acute angle and a side length. One leg alone can belong to many different right triangles.

Does the hypotenuse formula work for any triangle?
No. The Pythagorean theorem applies to right triangles. For a non-right triangle, the law of cosines is one method when the relevant angle and sides are known.

Do the two leg measurements need the same units?
Yes. Convert them to one unit before squaring and adding. The answer will then be expressed in that length unit.

How do I find a missing leg instead?
If the hypotenuse c and one leg a are known, the other leg is b = √(c² − a²). The hypotenuse must be longer than the known leg, and the measurements must use the same unit.

Why is my calculated hypotenuse longer than both legs?
That is expected for a right triangle with two positive leg lengths: its hypotenuse is the longest side. If it is not, revisit the inputs and calculation.

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