The Hypotenuse: What It Is and How to Find It

The hypotenuse is the longest side of a right triangle: the side opposite the 90° angle. To find it, square the two shorter sides (the legs), add the squares and take the square root: c = √a² + b². For legs 3 and 4, the hypotenuse is √9 + 16 = √25 = 5.

What is the hypotenuse?

Every right triangle has one 90° angle, and its three sides have special names. The two sides that meet at the right angle are the legs, usually called a and b. The third side, the one that does not touch the right angle, is the hypotenuse, usually called c.

The easiest way to find it in a diagram: put your finger on the little square that marks the right angle, then look straight across the triangle. The side you are looking at is the hypotenuse. This works however the triangle is turned.

Only right triangles have a hypotenuse. In an acute or obtuse triangle no side is called the hypotenuse, and the Pythagorean theorem formula does not apply as written.

Where the word comes from

"Hypotenuse" comes from the Greek hypoteinousa, meaning "stretching under". Greek geometers, including Euclid, described this side as the line that subtends (stretches across from) the right angle. The word passed into Latin as hypotenusa and then into English. So the name itself tells you where to look: the side stretched across from the right angle.

Three facts that are always true

• It is opposite the right angle. That is the definition, so it never changes.
• It is the longest side. Because c² = a² + b², the number c² is bigger than a² and bigger than b² (you add a positive amount to each). A bigger square means a longer side, so c beats both legs.
• It is shorter than the two legs added together. Going straight across is always shorter than going around the corner, so c < a + b.

There is a fourth, more surprising fact. If you draw the circle that passes through all three corners of a right triangle, the hypotenuse is a diameter of that circle. Turn it around and you get a way to build right triangles: pick any point on a semicircle and join it to the two ends of the diameter. The angle at that point is always 90° (this is Thales' theorem).

The hypotenuse formula

The Pythagorean theorem says a² + b² = c². Take the square root of both sides and you get the formula for the hypotenuse:

1. Square each leg. Multiply each leg by itself: a² = a × a. Squaring is not doubling, so 8² = 64, not 16.
2. Add the two squares. The total is c², the area of the square drawn on the hypotenuse.
3. Take the square root. Ask "what number times itself gives this?" That number is c. If the total is not a perfect square, leave the answer as a radical or round it.

Strictly, the equation c² = 289 has two solutions, 17 and −17. A length can't be negative, so you keep only the positive root. That's why the formula uses √ , which always means the positive square root.

How to find the hypotenuse: five worked examples

The method never changes; only the numbers do. These examples move from whole numbers to radicals, decimals, mixed units and a real-world problem.

For more practice, the hypotenuse worksheet has printable problems with an answer key, and the hypotenuse calculator checks your answers step by step.

Estimate before you calculate

A ten-second estimate catches most slips, such as forgetting the square root or typing a number wrong. Use these three checks:

• The hypotenuse is between the longer leg and the sum of the legs. For legs 9 and 40, c must be more than 40 and less than 49. (It is 41.)
• If the legs are equal, c is about 1.41 times a leg, because c = √2 a and √2 ≈ 1.414. This is the 45-45-90 triangle. Legs that are nearly equal behave almost the same way: for legs 20 and 21, expect about 1.41 × 20.5 ≈ 29. (It is exactly 29.)
• If one leg is much shorter than the other, c is only a little longer than the long leg. For legs 11 and 60, the hypotenuse is 61, just one more than the long leg.

If your answer is shorter than a leg, or longer than both legs added together, check your work before going on.

Finding the hypotenuse from one leg and an angle

Sometimes you know only one leg, plus one of the acute angles. The Pythagorean theorem needs two sides, so here you use trigonometry instead. In a right triangle with acute angle A:

Rearranging each one for c gives two hypotenuse formulas:

Make sure your calculator is in degree mode. Two special triangles skip the calculator entirely: in a 45-45-90 triangle the hypotenuse is a leg times √2, and in a 30-60-90 triangle it is exactly twice the shortest leg. To decide which tool a problem needs, see Pythagorean theorem or trigonometry?, or let the right triangle solver handle any combination of sides and angles.

Common errors when finding the hypotenuse

Almost every wrong answer comes from one of these five slips.

• Adding the legs. The hypotenuse is always less than a + b, as the comparison above shows.
• Stopping at c². If you get 25 for legs 3 and 4, you found the area of the square, not the side. Take the square root.
• Squaring the sum. (3 + 4)² = 49, but 3² + 4² = 25. Square each leg first, then add.
• Treating a known hypotenuse as a leg. If the problem gives you the hypotenuse and one leg, you are looking for a leg and must subtract: see how to find a missing side.
• Mixing units. Convert first, as in Example 4.

The full list, with a wrong-versus-right example for each, is on the common mistakes page.

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