This Pythagorean theorem quiz has 10 questions. Four test ideas: spotting the hypotenuse, choosing the correct equation, recognizing a triple and using the converse. Six test calculation: missing sides, rounding, a word problem, distance on a grid and a 3D diagonal. Each answer is checked instantly and comes with a worked solution.
How the quiz works
- Answer each question, then press Check answer. For multiple-choice questions, pick one option (arrow keys work too). For number questions, type your answer and press Enter or the button.
- Read the feedback. The question turns green or red, the correct option is marked, and a short explanation shows every step of the arithmetic, with a link to the lesson that teaches the skill.
- Watch your score. The score under the last question updates as you go. After all ten, you get a summary and a list of pages to review for anything you missed.
- Press New quiz for a fresh set of ten questions. The skills stay the same; the numbers, letters and answer order change.
You get one try per question, so the score reflects what you can do the first time, as on a test. That is deliberate: the explanation you see after a wrong answer is the most useful part of the quiz.
What you can type
Number boxes accept whole numbers (24), decimals (6.3), fractions (3/4) and square roots (5√2 or sqrt(50)). A unit at the end, as in 24 cm, is ignored. When a question asks you to round to the nearest tenth, any answer within 0.05 of the exact value counts, and so does the exact radical: for √40 ≈ 6.325, the answers 6.3, 6.32 and 2√10 are all marked correct.
Sharing a quiz
Every quiz has a number. After you press New quiz, the page address ends with that number, for example /pythagorean-theorem-quiz/?quiz=2024, and anyone who opens the link gets exactly the same ten questions. Teachers can use this to give a whole class one quiz and then go over the answers together.
What each question tests
The ten questions always appear in the same order, from the basic idea to the 3D extension, so a low score points straight at the skill to practice.
| # | Skill | What it checks | Learn it |
|---|---|---|---|
| 1 | Identify the hypotenuse | The hypotenuse is the side opposite the right angle, whatever the letters or the orientation. | Hypotenuse |
| 2 | Find the hypotenuse | Square both legs, add, take the square root. | Find the missing side |
| 3 | Choose the correct equation | Applying a2 + b2 = c2 when the sides have other names. | The formula |
| 4 | Find a missing leg | Subtracting the squares instead of adding them. | Find the missing side |
| 5 | Recognize a triple | Testing three whole numbers with a2 + b2 = c2. | Pythagorean triples |
| 6 | Round an irrational answer | Square roots of non-square numbers and rounding to a tenth. | Square roots |
| 7 | Classify with the converse | Comparing a2 + b2 with c2 to tell right, acute and obtuse apart. | The converse |
| 8 | Solve a word problem | Finding the right triangle in a ladder against a wall. | Word problems |
| 9 | Distance between two points | Turning coordinates into the legs of a right triangle. | Distance formula |
| 10 | Diagonal of a box | Using the theorem twice in three dimensions. | The theorem in 3D |
Questions 3 and 7 are the ones students most often get wrong when they rush: the first because the letters are not a, b and c, the second because the sides are listed out of order. The common mistakes page covers both traps.
Find the right angle first
Half of the quiz hinges on one decision: which side is the hypotenuse? Get it right and the rest is arithmetic. The reliable way is to find the right-angle mark first, then look straight across the triangle from it.
Three more habits catch most remaining errors:
- Add to find the hypotenuse, subtract to find a leg. The hypotenuse is the longest side, so its square is the biggest number in the equation. The step-by-step method shows both cases.
- Take the square root at the end. c2 = 100 is not the answer; c = 10 is.
- Check the size of your answer. A hypotenuse must be longer than either leg but shorter than the two legs added together. A leg must be shorter than the hypotenuse.
Printable quiz with answers
Prefer paper, or reading without JavaScript? Here is a second set of ten questions on the same skills. Work each one out, then open the solution to check.
1. In the triangle below, the square marks the right angle. Which side is the hypotenuse?
Show solution
KM. The right angle is at L, and the hypotenuse is the side that does not touch L. KL and LM are the legs. The bottom of the picture is not special: here the hypotenuse happens to be horizontal because the triangle is standing on it.
2. A right triangle has legs of 9 cm and 40 cm. How long is the hypotenuse?
Show solution
c2= 92 + 402 = 81 + 1,600 = 1,681c= √1,681 = 4141 cm. The sides 9, 40, 41 form a Pythagorean triple, so the answer is a whole number.
3. The hypotenuse of a right triangle is 25 m and one leg is 7 m. Find the other leg.
Show solution
b2= 252 − 72 = 625 − 49 = 576b= √576 = 2424 m. The hypotenuse is given, so you subtract.
4. A student says a right triangle with legs 5 and 7 has a hypotenuse of 12. What went wrong, and what is the correct length to the nearest tenth?
Show solution
The student added the lengths (5 + 7 = 12) instead of their squares. The two legs together are always longer than the hypotenuse, so 12 is too long.
c2= 52 + 72 = 25 + 49 = 74c= √74 ≈ 8.602About 8.6.
5. Which set is not a Pythagorean triple?
(A) 9, 12, 15 (B) 11, 60, 61 (C) 12, 18, 22 (D) 33, 56, 65
Show solution
(C). 122 + 182 = 144 + 324 = 468, but 222 = 484. The others work: 81 + 144 = 225 = 152, 121 + 3,600 = 3,721 = 612 and 1,089 + 3,136 = 4,225 = 652.
6. A square has sides of 8 inches. How long is its diagonal, to the nearest tenth of an inch?
Show solution
The diagonal splits the square into two right triangles with legs 8 and 8.
d2= 82 + 82 = 128d= √128 = 8√2 ≈ 11.314About 11.3 in. Shortcut: the diagonal of any square is side ×√2, the 45-45-90 triangle rule.
7. A triangle has sides 9, 12 and 7. Is it acute, right or obtuse?
Show solution
Use the longest side, 12, as c: 72 + 92 = 49 + 81 = 130 and 122 = 144. Because 130 < 144, the longest side is too long for a right angle. The triangle is obtuse. See acute, obtuse and right triangles.
8. A TV screen is 48 inches wide and 27 inches tall. TV sizes are measured along the diagonal. What size is this TV, to the nearest tenth of an inch?
Show solution
d2= 482 + 272 = 2,304 + 729 = 3,033d= √3,033 ≈ 55.073About 55.1 in, so it would be sold as a 55-inch TV.
9. Find the distance between A(1, 1) and B(7, 9).
Show solution
Draw a right triangle under the segment. The horizontal leg is 7 − 1 = 6 and the vertical leg is 9 − 1 = 8.
d = √62 + 82 = √36 + 64 = √100 = 1010 units. This is exactly what the distance formula computes.
10. A box is 12 in long, 4 in wide and 3 in tall. How long is the diagonal from one bottom corner to the opposite top corner?
Show solution
First the diagonal of the base, then use it as a leg with the height:
e2= 122 + 42 = 144 + 16 = 160d2= 160 + 32 = 169d= √169 = 1313 in. In one step: d = √122 + 42 + 32. More on the Pythagorean theorem in 3D.
What your score means
The quiz is short, so treat your score as a signal rather than a grade. Use it to decide what to do next:
- 9 or 10: you have the core skills. Move on to word problems, the challenge problems or SAT and ACT questions.
- 6 to 8: read the explanation for each miss, then take a new quiz. If the same skill keeps tripping you up, the worked examples cover every type of question here.
- 5 or fewer: start with what the Pythagorean theorem says and how to use it step by step, then practice on the worksheet before trying the quiz again.
The questions follow the skills in the Common Core grade 8 standards, so the quiz works as a quick check at the end of a unit. The 8th grade standards page maps each skill to its standard.
Frequently asked questions
How many questions are on the Pythagorean theorem quiz?
Ten. Four are multiple-choice questions about ideas (the hypotenuse, the equation, triples and the converse) and six ask you to calculate a length, including one rounded answer, one word problem, one coordinate distance and one 3D diagonal.
Can I retake the quiz?
Yes, as often as you like. Press New quiz and you get ten new questions on the same skills with different numbers, letters and answer orders, so you can't simply remember the answers from last time.
Do I have to round my answers?
Only when the question says so. Most answers are whole numbers because the questions use Pythagorean triples. For the rounding question, give the answer to the nearest tenth or type the exact radical, such as 2√10; both are accepted.
Can a teacher give the whole class the same quiz?
Yes. Each quiz has a number, shown above the first question and stored in the page address after you press New quiz. Share that link and every student gets identical questions, which makes it easy to review the answers together.
Is this quiz right for 8th grade?
Yes. It covers what the grade 8 standards expect: finding unknown sides in two and three dimensions, using the converse, and finding the distance between two points. Questions on proofs are better handled in class discussion than in a multiple-choice quiz.
Keep exploring
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- PracticePythagorean Theorem WorksheetsFree Pythagorean theorem worksheets with answers: a printable worksheet generator plus ready-made sheets on the hypotenuse, missing sides and word problems.
- LearnCommon Pythagorean Theorem MistakesTwelve common Pythagorean theorem mistakes, from a + b = c to misusing the converse, each with a wrong-vs-right worked example and a final self-check list.
- PracticePythagorean Theorem ExamplesFifteen Pythagorean theorem examples solved step by step, from finding the hypotenuse to 3D diagonals, coordinates and algebra, each with a diagram and answer.
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- PracticePythagorean Theorem Challenge Problems12 hard Pythagorean theorem challenge problems: an ant on a box, folded paper, sliding and crossed ladders, Dudeney's spider, with hints and full solutions.