Pythagorean Theorem Worksheet Generator

A Pythagorean theorem worksheet generator makes printable practice sheets for finding side lengths in right triangles, with an answer key on a separate page. It runs in your web browser, is free, and needs no sign-up. The problems are made on your own device, and nothing is uploaded.

You can include five problem types: finding the hypotenuse, finding a missing leg, checking whether a triangle is a right triangle, finding the distance between two points, and solving word problems. Choose how many problems you want (6 to 30) and whether they are Easy, Medium or Hard. Teachers can print a set for a class, and students can save a practice sheet as a PDF handout. The converse problems test the Pythagorean relationship between the three sides,. Everything you make is free for commercial use, such as on Teachers Pay Teachers.

How to use the Pythagorean theorem worksheet generator

1. Tick one or more problem types under Problem types. The page opens with Find the hypotenuse and Find a missing leg already ticked.
2. Choose a Difficulty: Easy, Medium or Hard.
3. Pick the Number of problems, from 6 to 30.
4. Open More options if you want to change the Columns (Automatic, 1, 2 or 3), the Paper size (Letter or A4), add a Worksheet title, show a Name and date line, or turn off the answer key on a separate page (it is on by default).
5. Check the preview on the right. It shows the sheet exactly as it will print.
6. Click Show answers to see each answer under its problem in the preview. Press it again to hide them.
7. Click Print / Save as PDF. In the print dialog, choose Save as PDF if you want a file instead of paper.

Click New worksheet for a fresh set of random problems with the same settings. Click Copy link to copy a link that opens exactly the same worksheet again. Your settings are remembered in this browser for your next visit, unless you open a shared link, which uses that link’s settings. When several problem types are ticked, the problems are shared out evenly and grouped by type, each group with its own instruction, and numbered 1 to N across the sheet.

What is the Pythagorean theorem formula

The Pythagorean theorem formula is a² + b² = c². In a right triangle, a and b are the two legs, the sides that meet at the right angle. The hypotenuse, c, is the side opposite the right angle, and it is always the longest side.

To find the hypotenuse, use c = √(a² + b²). Square each leg, add the results, and take the square root. For legs 8 and 15, 8² + 15² = 64 + 225 = 289, and √289 = 17.

Easy problems use Pythagorean triples such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25, and their multiples. These always give whole-number answers. Other answers are written as a simplified radical and rounded to the nearest tenth. For legs 20 and 18, c = √724 = 2√181 ≈ 26.9.

How do you find a missing leg

To find a missing leg, rearrange the formula to b = √(c² − a²), where c is the hypotenuse and a is the known leg. Subtracting the known leg’s square from the hypotenuse’s square gives the square of the missing leg.

1. Square the hypotenuse.
2. Square the known leg.
3. Subtract the known leg’s square from the hypotenuse’s square.
4. Take the square root of the result.

For a hypotenuse of 17 and a leg of 8: 17² − 8² = 289 − 64 = 225, and √225 = 15. Check it by putting the sides back into the formula: 8² + 15² = 64 + 225 = 289 = 17².

The leg is always shorter than the hypotenuse, so the subtraction always gives a positive number. For a hypotenuse of 29 and a leg of 25, 29² − 25² = 841 − 625 = 216, so the missing leg is √216 = 6√6 ≈ 14.7.

Sample problems from this generator

1. The legs of a right triangle are 6 and 2. Find the hypotenuse x. Answer: x = √40 = 2√10 ≈ 6.3
2. The legs of a right triangle are 3 and 6. Find the hypotenuse x. Answer: x = √45 = 3√5 ≈ 6.7
3. The hypotenuse of a right triangle is 12 and one leg is 3. Find the other leg x. Answer: x = √135 = 3√15 ≈ 11.6
4. The hypotenuse of a right triangle is 15 and one leg is 2. Find the other leg x. Answer: x = √221 ≈ 14.9

Frequently asked questions

How do you know if a triangle is a right triangle?

Use the converse of the Pythagorean theorem. Take the longest side as c and check whether a² + b² = c². If it does, the triangle is a right triangle. For sides 60, 63 and 87, 60² + 63² = 3600 + 3969 = 7569 = 87², so it is a right triangle.

Is there an answer key?

Yes. The answer key is on a separate page by default and starts after the worksheet when you print. It lists every problem number with its answer, giving the exact value and the value rounded to the nearest tenth when the answer is not a whole number.

Does the generator include word problems?

Yes. The word problems type covers situations such as ladders, fields, boats, ramps and screens, and the answers include the unit. For example, a boat that sails 14 miles south and then 35 miles west is √1421 ≈ 37.7 miles from its start.

Can I make several versions of the same worksheet?

Yes. Each click on New worksheet creates different problems with the same settings. Teachers can print one version per row of desks, and students can practise again with new numbers.

Does it work on a phone or tablet?

Yes, it works on phones and tablets as well as computers. You need a printer to print on paper, or you can choose Save as PDF in the print dialog.

Are the Pythagorean theorem worksheets I make here free for commercial use?

Yes. Everything you make here is royalty-free for personal, classroom and commercial use, including Teachers Pay Teachers, Etsy printables, KDP workbooks and print-on-demand products. No credit is required. The only thing you may not do is copy the generator or the site itself to run a competing service. See the terms.

Related tools

• Simplifying radicals worksheets: use these when you need to simplify answers such as √724 into 2√181.
• Exponents worksheets: use these to practise squaring and square roots before you work with the formula.
• Slope worksheets: use these to practise coordinate-plane skills that support the distance problems.
• Angles worksheets: use these to practise the angle facts that go with triangles, including the right angle.
• Transformations worksheets: use these to practise moving shapes on a coordinate grid.
• Algebra worksheets: browse the full set of worksheet tools on this site.
• Special right triangles worksheets: use these for 45-45-90 and 30-60-90 triangles in simplest radical form.
• Volume and surface area worksheets: use these for cylinders, cones and pyramids, where the slant height often needs the Pythagorean theorem.