Finding the Hypotenuse Worksheet

This finding the hypotenuse worksheet has 16 right triangles in which both legs are known and the longest side is missing. Students square the two legs, add them and take the square root: c = √a² + b². Print it with the button below. The answer key, with every step shown, starts on a new page.

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What the worksheet practices

The hypotenuse is the side across from the right angle, and it is always the longest side of a right triangle. Every problem gives the two legs, so every problem uses the same move: add the squares. The difficulty climbs in four steps.

• Problems 1–8, whole-number answers. The legs come from Pythagorean triples such as 3-4-5, 8-15-17 and 20-21-29, so the square root always comes out exact. Problem 7 names the sides with vertex letters: find AB in triangle ABC.
• Problems 9–12, radical answers. Here a² + b² is not a perfect square. Students give the exact answer in simplest radical form, such as √45 = 3√5, and a decimal rounded to the nearest hundredth. Problem 12 uses triangle PQR with the right angle at Q.
• Problems 13–14, decimal legs (1.5 and 2; 2.4 and 7) that still give exact decimal answers.
• Problems 15–16, mixed review: one answer that cannot be simplified (√149) and one decimal answer that must be rounded.

The triangles are turned and flipped on purpose. A student who thinks of the hypotenuse as "the slanted side on the right" has to look for the right-angle mark instead, and the side opposite that mark is the hypotenuse every time. Figures are not drawn to scale; a few very long, thin triangles are drawn less extreme than their true shape so the labels stay readable.

How to use it

• Before: students should know the perfect squares up to about 30² = 900 and how to simplify a square root. The square roots and radicals page reviews both.
• Calculators: problems 1–8 work well without one. Allow a calculator for 9–16, or ask for exact radical answers only.
• Self-checking: students can check single answers with the hypotenuse calculator, which shows the same steps as the answer key.
• Time: plan on roughly 25–35 minutes for the full sheet, or assign the odd problems for a shorter set.

Teacher notes

• Watch for (a + b)². The most common error is adding the legs before squaring. On problem 1 that gives √(3 + 4)² = 7 instead of 5. A quick test catches it: the hypotenuse must be shorter than the two legs added together. More slips like this are on the common mistakes page.
• Spot the multiples. Problems 2 and 4 are the 3-4-5 triangle scaled by 2 and 3, problem 8 is the 5-12-13 triangle doubled, and problem 13 is 3-4-5 halved. Students who notice can skip the arithmetic.
• Problem 10 previews a shortcut. Equal legs of 4 give 4√2; in any isosceles right triangle the hypotenuse is a leg times √2, the rule behind the 45-45-90 triangle.
• Units: the answer has the same unit as the legs. Problems with units are 3, 5, 6, 8, 13, 14 and 16.
• Standard: Common Core 8.G.B.7, applying the Pythagorean theorem to find unknown side lengths in right triangles.

Ready for the next step? The missing side worksheet mixes in problems where a leg is missing, and the worksheet generator builds as many new hypotenuse sheets as you need, each with its own answer key.