The Pythagorean theorem says that in a right triangle the squares of the two legs add up to the square of the hypotenuse: a2 + b2 = c2. This cheat sheet puts everything you need in one place: the formula and its rearrangements, the converse, special right triangles, common triples, the distance and 3D formulas, and the perfect squares from 1 to 25.
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The formula and its rearrangements
The Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c (opposite the right angle), a2 + b2 = c2.
Rearranged: c = √a2 + b2
a = √c2 − b2 and b = √c2 − a2
Converse and special right triangles
Converse (c longest, a + b > c): a2 + b2 = c2 right (5, 12, 13); > acute (5, 6, 7); < obtuse (5, 6, 9).
Distance, circles and 3D
Box diagonal: d = √l2 + w2 + h2
3D distance: √Δx2 + Δy2 + Δz2; cube diagonal s√3
Pythagorean triples and perfect squares
All 16 primitive Pythagorean triples with c ≤ 100 (no common factor):
| 3, 4, 5 | 5, 12, 13 | 8, 15, 17 | 7, 24, 25 |
| 20, 21, 29 | 12, 35, 37 | 9, 40, 41 | 28, 45, 53 |
| 11, 60, 61 | 16, 63, 65 | 33, 56, 65 | 48, 55, 73 |
| 13, 84, 85 | 36, 77, 85 | 39, 80, 89 | 65, 72, 97 |
Multiples are triples too: 6-8-10, 9-12-15, 12-16-20, 15-20-25, 10-24-26, 16-30-34.
Euclid's formula: whole numbers m > n > 0 give the triple m2 − n2, 2mn, m2 + n2.
Perfect squares from 1 to 25
| 12 = 1 | 22 = 4 | 32 = 9 | 42 = 16 | 52 = 25 |
| 62 = 36 | 72 = 49 | 82 = 64 | 92 = 81 | 102 = 100 |
| 112 = 121 | 122 = 144 | 132 = 169 | 142 = 196 | 152 = 225 |
| 162 = 256 | 172 = 289 | 182 = 324 | 192 = 361 | 202 = 400 |
| 212 = 441 | 222 = 484 | 232 = 529 | 242 = 576 | 252 = 625 |
Useful square roots: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449, √7 ≈ 2.646, √10 ≈ 3.162.
Simplifying a root: pull out the largest square factor: √50 = √25 · 2 = 5√2, √72 = 6√2, √12 = 2√3.
Steps, checks and related formulas
Four steps: (1) label c, the side opposite the right angle; (2) substitute into a2 + b2 = c2; (3) add the squares to find c, subtract to find a leg; (4) take the square root and round only if asked.
Quick checks: the hypotenuse is longer than each leg but shorter than the two legs added; c2 is not the answer, c is; no right angle means you need the law of cosines.
Related facts: area = 1 over 2ab; height to the hypotenuse = ab over c; sinA = a over c, cosA = b over c, tanA = a over b; the Pythagorean identity sin2θ + cos2θ = 1; the law of cosines c2 = a2 + b2 − 2abcosC.
How to use this cheat sheet
Each block above condenses a full lesson. Keep the sheet next to your homework at first, then try a worksheet without it; the goal is to need it less each time. The four steps, the first four triples and the special triangle ratios are worth memorizing, because the ACT gives you no formula sheet and the SAT reference sheet lists only the theorem and the special triangles (see the theorem on the SAT and ACT).
The perfect squares table does double duty. Reading it forward gives squares for the substitution step; reading it backward tells you at a glance whether a sum such as 169 has a whole-number square root. If a number isn't in the table, the answer is irrational: leave it as a simplified root or round it as the question asks. For the full explanation of any line, start from the formula page or look up a term in the glossary.
Frequently asked questions
What is the Pythagorean theorem formula?
a2 + b2 = c2, where a and b are the legs of a right triangle and c is the hypotenuse, the side opposite the right angle. Rearranged, c = √a2 + b2 and a = √c2 − b2.
Which Pythagorean triples should I memorize?
Start with 3-4-5, 5-12-13, 8-15-17 and 7-24-25, and learn to spot their multiples such as 6-8-10 and 10-24-26. Those cover most textbook and test problems with whole-number answers.
Can I use this cheat sheet on a test?
That depends on your teacher or the exam. The SAT gives you its own reference sheet with the theorem and the special right triangles; the ACT provides no formulas. Use this page to study, and check the rules before any test.
How do I print the cheat sheet?
Press the Print button at the top of the page, or use your browser's print command. The menus, buttons, section headings and this FAQ are left out automatically, so the reference fits on two pages. Turn on background graphics to keep the colored diagrams.
Does the Pythagorean theorem work for every triangle?
No, only for right triangles. For other triangles, the law of cosines adds a correction term, and comparing a2 + b2 with c2 tells you whether the largest angle is acute or obtuse.
Keep exploring
- LearnThe Pythagorean Theorem FormulaThe Pythagorean theorem formula a² + b² = c²: what each letter means, the algebra to solve for any side, units, coordinate and 3D forms, and worked examples.
- TriplesList of Pythagorean TriplesComplete, computer-checked list of Pythagorean triples: all 52 with hypotenuse up to 100 and all 47 primitive triples up to 300, with m, n, perimeter and area.
- 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.
- CalculatorsPythagorean Theorem CalculatorFree Pythagorean theorem calculator. Enter any two sides of a right triangle to find the third, with step-by-step working, exact radical answers and a drawing.
- LearnPythagorean Theorem GlossaryPythagorean theorem vocabulary from A to Z: over 50 terms such as leg, hypotenuse, converse and primitive triple, each clearly defined and linked to a lesson.
- PracticePythagorean Theorem WorksheetsFree Pythagorean theorem worksheets with answers: a printable worksheet generator plus ready-made sheets on the hypotenuse, missing sides and word problems.