The Pythagorean Theorem Is the Standard Method

To find the hypotenuse of a right triangle, use the Pythagorean theorem: a² + b² = c², where a and b are the two shorter sides and c is the hypotenuse (the longest side, opposite the right angle). Square each of the two shorter sides, add those squares together, then take the square root of the result. That final number is your hypotenuse.

The theorem works for any right triangle, regardless of size. You need only the lengths of the two sides that form the right angle — the sides that meet at the 90-degree corner.

Key Takeaways

  • The Pythagorean theorem (a² + b² = c²) is the direct way to find a hypotenuse when you know the two shorter sides.
  • Square both shorter sides separately, add those results, then take the square root of the sum to get the hypotenuse length.
  • A calculator with a square root function makes the math faster, though the method works on paper with long division.
  • The hypotenuse is always the longest side of a right triangle and always sits across from the right angle.
  • If you know the hypotenuse and one other side, you can rearrange the formula to find the missing side instead.

Step-by-Step Calculation

Start with the two sides that meet at the right angle. Call one side a and the other side b. Multiply a by itself to get a². Do the same for b to get b². For example, if one side is 3 units and the other is 4 units, then a² = 9 and b² = 16.

Add a² and b² together. In the example above, 9 + 16 = 25. Now find the square root of that sum. The square root of 25 is 5, so the hypotenuse is 5 units. You can check this: 3² + 4² = 9 + 16 = 25, and √25 = 5.

If your sum does not have a clean square root, use a calculator. For instance, if the two sides are 5 and 7, then 5² + 7² = 25 + 49 = 74. The square root of 74 is approximately 8.6, so the hypotenuse is about 8.6 units.

Using a Calculator or Spreadsheet

Most scientific calculators have a square root button (√). Enter your sum and press it to get the result when ready. On a phone calculator, switch to scientific mode to access the square root function.

In a spreadsheet like Excel or Google Sheets, use the SQRT function. Type =SQRT(a2+b2) in a cell, where a2 and b2 are the cells containing your two side lengths. The spreadsheet will calculate the hypotenuse automatically. This method is useful if you need to find the hypotenuse for many triangles at once.

When You Know the Hypotenuse but Not One Side

If you have the hypotenuse and one of the shorter sides, rearrange the formula to find the missing side. The formula becomes a² = c² − b², where c is the hypotenuse and b is the known side. Square the hypotenuse, subtract the square of the known side, then take the square root of the result.

For example, if the hypotenuse is 10 and one side is 6, then a² = 10² − 6² = 100 − 36 = 64. The square root of 64 is 8, so the missing side is 8 units. Check: 6² + 8² = 36 + 64 = 100 = 10².

Common Right Triangle Ratios

Some right triangles have sides in ratios you may recognize without calculating. The 3-4-5 triangle is the most famous: sides of 3 and 4 always produce a hypotenuse of 5. Multiples of this ratio also work — a 6-8-10 triangle, a 9-12-15 triangle, and so on.

Other common ratios include 5-12-13 and 8-15-17. If your triangle matches one of these patterns, you can state the hypotenuse without a calculator. For most other combinations, the Pythagorean theorem and a calculator are your fastest route.

Avoiding Common Mistakes

The most frequent error is forgetting to take the square root at the end. If you stop after adding a² and b², you have the square of the hypotenuse, not the hypotenuse itself. Always complete the final step.

Another mistake is using the wrong sides. The Pythagorean theorem applies only to right triangles, and only when a and b are the two sides that form the right angle. If you use the hypotenuse as one of your input values, your answer will be wrong. Double-check that you are squaring the two shorter sides, not the longest one.

Frequently Asked Questions

What if the triangle is not a right triangle?

The Pythagorean theorem does not work for triangles without a 90-degree angle. For other triangles, you need different formulas, such as the law of cosines, which requires knowing an angle as well as the sides.

Can the hypotenuse be shorter than one of the other sides?

No. The hypotenuse is always the longest side of a right triangle. If your calculation produces a hypotenuse shorter than one of your input sides, you made an error — most likely using the wrong sides or forgetting the square root step.

How many decimal places should I round to?

That depends on your purpose. For rough estimates, one or two decimal places is usually enough. For construction or engineering, match the precision of your input measurements. If your sides are measured to the nearest inch, rounding your hypotenuse to the nearest inch makes sense.

Do I need a calculator, or can I do this by hand?

You can do it by hand if the square root is a whole number. For messier results, a calculator saves time and reduces errors. Even a basic four-function calculator with a square root button works fine.