The best way to practice the Pythagorean theorem is in order of difficulty: first find hypotenuses, then missing legs and radicals, then word problems and the converse, then coordinates and 3D, and finally challenge problems. This page lays out that five-level ladder, gives a warm-up question for each level, and links every practice resource on the site.
Not sure where you stand? Do the five warm-up problems first, then start at the first level whose problem you got wrong. If the theorem itself is new to you, begin with the lesson What is the Pythagorean theorem?
The practice ladder: five levels
Each level builds on the one before. Move up when you can solve that level's problems correctly without looking anything up, not just when you've read the examples.
Level 1: find the hypotenuse
Given two legs, find the longest side with c = √a2 + b2. You should be able to spot the hypotenuse in any orientation and get whole-number answers like 9-40-41 quickly.
- Learn it: finding the hypotenuse
- See it done: worked examples 1–3
- Practice: hypotenuse worksheet and practice problems 1–2
Level 2: missing legs, radicals and decimals
Subtract to find a leg, give exact answers in simplest radical form such as 2√10, and handle decimals and mixed units.
- Learn it: finding a missing side and square roots and radicals
- Practice: missing side worksheet, practice problems 3–8, examples 4–9
- Check your work: the Pythagorean theorem calculator shows every step
Level 3: word problems and the converse
Turn ladders, ramps and screens into labeled right triangles, and use the converse to decide whether three lengths make a right, acute or obtuse triangle.
- Practice: 15 word problems with a four-step strategy, the word problems worksheet and the converse worksheet
- Test yourself: the Pythagorean theorem quiz
Level 4: coordinates, 3D and multi-step problems
Find distances between points, diagonals of boxes, heights of isosceles triangles and trapezoids, and sides written with a variable.
- Practice: distance formula worksheet, 3D worksheet, practice problems 11–30
- Test prep: SAT and ACT questions and the cheat sheet
Level 5: challenge problems and proof
Problems where you must create the right triangle yourself: shortest paths on a box, folds, sliding and crossed ladders. Proving the theorem belongs here too.
- Practice: 12 challenge problems with hints
- Go deeper: proofs of the theorem
Five warm-up problems
One problem per level. Work each on paper, then open the answer. The first one you miss tells you which level to start on.
Warm-up 1 Level 1
The legs of a right triangle are 15 and 36. How long is the hypotenuse?
Show answer
c = √152 + 362 = √225 + 1296 = √1521 = 3939. (It's the 5-12-13 triangle multiplied by 3.)
Warm-up 2 Level 2
A right triangle has hypotenuse 7 and one leg 3. Find the other leg in simplest radical form.
Show answer
b = √72 − 32 = √49 − 9 = √40 = 2√10 ≈ 6.322√10. Subtract because the hypotenuse is given, then take out the square factor: 40 = 4 × 10.
Warm-up 3 Level 3
A carpenter builds a rectangular frame 20 in by 48 in and measures its diagonal as 52 in. Are the corners square (exactly 90°)?
Show answer
202 + 482 = 400 + 2304 = 2704 = 522Yes. By the converse of the theorem, the squares match, so the corner is a right angle. Builders use exactly this check.
Warm-up 4 Level 4
A box measures 8 cm by 4 cm by 1 cm. How long is the space diagonal, from one corner to the opposite corner through the inside?
Show answer
d = √82 + 42 + 12 = √64 + 16 + 1 = √81 = 99 cm. The 3D version of the theorem adds a third square.
Warm-up 5 Level 5
The legs of a right triangle differ by 7, and the hypotenuse is 17. Find the legs.
Show answer
x2 + (x + 7)2= 1722x2 + 14x + 49= 289x2 + 7x − 120= 0(x − 8)(x + 15)= 08 and 15. Reject x = −15 because lengths are positive. If you solved this, the challenge problems are ready for you.
Types of Pythagorean theorem problems
Almost every question you will meet is one of these types. Knowing the type tells you the first move.
| Problem type | You are given | First move | Practice |
|---|---|---|---|
| Find the hypotenuse | two legs | add the squares: c = √a2 + b2 | worksheet |
| Find a leg | hypotenuse and one leg | subtract: a = √c2 − b2 | worksheet |
| Exact answers | a sum that isn't a perfect square | simplify the radical | simplifier |
| Right triangle or not | three lengths | compare a2 + b2 with c2 | worksheet |
| Acute or obtuse | three lengths | is a2 + b2 bigger or smaller than c2? | lesson |
| Special right triangles | 45-45-90 or 30-60-90 shapes | use the side ratios | 45-45-90, 30-60-90 |
| Hidden right triangles | isosceles triangles, rectangles, rhombuses | draw the height or diagonal | examples 8–10 |
| Distance on a grid | two points | legs are Δx and Δy | worksheet |
| 3D diagonals | boxes, cubes, pyramids | d = √a2 + b2 + c2 | worksheet |
| Word problems | a real situation | draw, label, solve, sense-check | 15 problems |
| Algebraic sides | sides like x, x + 7, x + 8 | substitute, expand, solve | problems 18–22 |
| Multi-step and puzzles | a figure with no obvious triangle | add a line to create one | challenges |
How to practice effectively
How you practice matters as much as how much. A large review of study techniques (Dunlosky and colleagues, Psychological Science in the Public Interest, 2013) rated two methods most useful: testing yourself and spreading practice out over time. Here is what that means for this topic.
- Try before you look. Attempt every problem before opening the solution, even if you are unsure. Struggling to recall a method is what makes it stick.
- Space it out. Three 20-minute sessions on different days usually beat one hour-long session. Come back to a level a few days after you "finished" it.
- Mix problem types. Once you know two or more types, practice them shuffled. Real tests don't tell you whether to add or subtract, so practice choosing.
- Keep an error log. For every mistake, write down what went wrong and the fix. Patterns appear quickly: the same slip tends to come back until you name it.
- Sense-check every answer. The hypotenuse must be the longest side and shorter than the two legs combined. That one check catches most errors.
A sample error log:
| Problem | What went wrong | Fix for next time |
|---|---|---|
| Leg with hypotenuse 26 | added the squares, got a "leg" longer than 26 | hypotenuse given → subtract |
| Isosceles triangle height | used the whole base as a leg | the height splits the base in half |
| Distance (−1, 4) to (4, −2) | wrote 4 − 1 = 3 for Δx | subtracting a negative adds: 4 − (−1) = 5 |
| Two-step problem | rounded √13 to 3.6 too early | keep radicals exact until the end |
The common mistakes page explains each of these slips in detail.
A one-week practice plan
If you have a test coming up, this plan covers every level in about 20 to 30 minutes a day and builds in review:
| Day | Do this |
|---|---|
| 1 | The warm-ups above, then worked examples 1–7 |
| 2 | Practice problems 1–10; start your error log |
| 3 | Word problems 1–8 |
| 4 | Practice problems 11–20, then redo yesterday's mistakes |
| 5 | Word problems 9–15 and a generated worksheet of mixed types |
| 6 | Practice problems 21–30, or the SAT and ACT set |
| 7 | The quiz, timing yourself, then everything in your error log once more |
Every practice resource
All the practice pages on the site, from worked examples to printable worksheets:
- 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.
- PracticePythagorean Theorem Word Problems15 Pythagorean theorem word problems from real life (ladders, TVs, ramps, ships, kites, roofs) with a 4-step strategy, a sketch for each and full solutions.
- PracticePythagorean Theorem Practice Problems30 Pythagorean theorem practice problems in three levels: missing sides, radicals, the converse, coordinates and algebra, each with a worked solution.
- 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.
- PracticePythagorean Theorem QuizTake a free Pythagorean theorem quiz: 10 questions on the hypotenuse, missing sides, triples, the converse and distance, with instant worked solutions.
- PracticePythagorean Theorem WorksheetsFree Pythagorean theorem worksheets with answers: a printable worksheet generator plus ready-made sheets on the hypotenuse, missing sides and word problems.
- PracticeFinding the Hypotenuse WorksheetFree printable finding the hypotenuse worksheet: 16 right triangles in varied orientations, from 3-4-5 triples to radicals and decimals, with worked answers.
- PracticeFind the Missing Side WorksheetPrintable find the missing side worksheet: 16 Pythagorean theorem problems to find a missing leg or the hypotenuse, with a step-by-step answer key.
- PracticePythagorean Theorem Word Problems WorksheetPrintable Pythagorean theorem word problems worksheet: 12 real-world problems with room to work, plus an answer key with labeled sketches and full solutions.
- PracticeConverse of the Pythagorean Theorem WorksheetConverse of the Pythagorean theorem worksheet: 16 printable problems on right, acute and obtuse triangles from their sides, with every step in the answer key.
- PracticeDistance Formula WorksheetPrintable distance formula worksheet: 12 Pythagorean theorem coordinate plane problems on grids, from whole numbers to radicals, with a worked answer key.
- Practice3D Pythagorean Theorem WorksheetPrintable 3D Pythagorean theorem worksheet: 12 problems on boxes, cubes, cones and square pyramids, each with a 3D drawing, plus a fully worked answer key.
- PracticeThe Pythagorean Theorem on the SAT and ACTHow the Pythagorean theorem appears on the SAT and ACT, what each test gives you, the triple and special-triangle shortcuts, and 10 original practice questions.
- PracticePythagorean Theorem Cheat SheetPrintable Pythagorean theorem cheat sheet: the formula and its rearrangements, the converse, special right triangles, triples, distance formulas and squares.
Frequently asked questions
What is the best way to practice the Pythagorean theorem?
Work in order of difficulty, attempt each problem before checking the answer, and mix problem types once you know them. Short, spaced sessions with an error log work better than long cramming sessions.
Where can I find Pythagorean theorem practice problems with answers?
Every problem set on this site has full solutions: 30 practice problems, 15 word problems, 12 challenge problems, and printable worksheets with answer keys on a separate page.
What should I know before practicing?
You need squares and square roots, and you need to recognize a right angle. If those feel shaky, review square roots and radicals and the parts of a right triangle first.
Are there printable Pythagorean theorem worksheets?
Yes. There are printable worksheets for the hypotenuse, missing sides, word problems, the converse, the distance formula and 3D problems, plus a worksheet generator that makes new sets with an answer key.
How do I prepare for a test on the Pythagorean theorem?
Follow the one-week plan above, take the quiz, and review the cheat sheet the night before. Redo every problem in your error log without looking at the solutions.
Keep exploring
- PracticePythagorean Theorem Practice Problems30 Pythagorean theorem practice problems in three levels: missing sides, radicals, the converse, coordinates and algebra, each with a worked solution.
- 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.
- PracticePythagorean Theorem WorksheetsFree Pythagorean theorem worksheets with answers: a printable worksheet generator plus ready-made sheets on the hypotenuse, missing sides and word problems.
- PracticePythagorean Theorem QuizTake a free Pythagorean theorem quiz: 10 questions on the hypotenuse, missing sides, triples, the converse and distance, with instant worked solutions.
- LearnLearn the Pythagorean TheoremLearn the Pythagorean theorem step by step: a free course in four levels, from a² + b² = c² and right triangles to distance, 3D and trig, with a self-check.
- PracticePythagorean Theorem Cheat SheetPrintable Pythagorean theorem cheat sheet: the formula and its rearrangements, the converse, special right triangles, triples, distance formulas and squares.