This course teaches the Pythagorean theorem, a2 + b2 = c2, from the ground up. Level 1 explains what the theorem says and how to use it. Level 2 applies it to coordinates, 3D and special triangles, Level 3 connects it to trigonometry, and Level 4 follows it beyond school geometry. Each lesson is a single page with diagrams, worked examples and answers.
Most students need only Level 1 for a first unit on the theorem. After each level, test yourself on the practice pages before moving on.
Before you start: prerequisites
The theorem rests on a few ideas. If one feels shaky, review it first.
- Squares: 42 = 4 × 4 = 16. Knowing the perfect squares up to 152 = 225 saves time.
- Square roots: √16 = 4 because 42 = 16. Most roots are not whole numbers: √50 ≈ 7.07 lies between 7 and 8 because 49 < 50 < 64. See square roots and radicals.
- Right angles: exactly 90°, a square corner, marked with a small square.
- One-step equations: solving x + 16 = 25 by subtracting 16 from both sides.
New words are defined in the glossary.
Where should I start? A five-question check
Answer each question on paper before opening it, then use the guide below.
1. Work out 122 and √64.
122 = 12 × 12 = 144 and √64 = 8, because 8 × 8 = 64.
2. In a right triangle, which side is the hypotenuse?
The side opposite (across from) the right angle. It does not touch the right angle, and it is always the longest side.
3. A right triangle has legs 8 and 15. How long is the hypotenuse?
c = √82 + 152 = √64 + 225 = √289 = 174. The hypotenuse of a right triangle is 10 and one leg is 6. How long is the other leg?
b = √102 − 62 = √100 − 36 = √64 = 8To find a leg you subtract, not add.
5. Can 6, 7 and 9 be the side lengths of a right triangle?
No. The two shorter sides give 62 + 72 = 36 + 49 = 85, but the longest gives 92 = 81. The two results are not equal, so the triangle has no right angle.
How to use your result:
- Question 1 was hard: review the prerequisites above and the square roots lesson, then start Level 1 at the beginning.
- Question 2 or 3 was hard: start Level 1 at the beginning, with what the Pythagorean theorem is.
- Only question 4 was hard: go straight to finding a missing side, then the step-by-step method.
- Only question 5 was hard: read the converse of the theorem.
- All five correct: skim common mistakes, then start Level 2.
Level 1: Foundations
Level 1 What the theorem says, why it needs a right angle, and how to find any missing side. If radicals are new to you, do square roots before the hypotenuse and missing-side lessons.
| Lesson | Key formula or idea |
|---|---|
| What is the theorem? | a2 + b2 = c2 in every right triangle |
| The formula | c = √a2 + b2, a = √c2 − b2 |
| How to use it | Label, square, add or subtract, square root, check |
| Right triangles | The hypotenuse is opposite the right angle; ∠A + ∠B = 90° |
| Find the hypotenuse | Add the squares: c = √a2 + b2 |
| Find a missing leg | Subtract the squares: a = √c2 − b2 |
| Square roots and radicals | √50 = √25 · 2 = 5√2 |
| The converse | If a2 + b2 = c2, the angle opposite c is 90° |
| Common mistakes | √a2 + b2 ≠ a + b |
- LearnWhat Is the Pythagorean Theorem?What the Pythagorean theorem is: a plain-English definition, the formal statement a² + b² = c², what it means with squares, and why it needs a right angle.
- 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.
- LearnHow to Do the Pythagorean Theorem, Step by StepHow to do the Pythagorean theorem in 8 steps: label the sides, add or subtract the squares, take the square root and check. Flowchart and worked examples.
- LearnRight Triangle: Parts, Legs and HypotenuseWhat a right triangle is, the names of its parts, how to spot the hypotenuse in any position, why it is always the longest side, and area and perimeter.
- LearnThe Hypotenuse: What It Is and How to Find ItWhat the hypotenuse of a right triangle is, why it is always the longest side, and how to find it with c = √(a² + b²): five worked examples plus a trig method.
- LearnHow to Find the Missing Side of a Right TriangleHow to find a missing leg of a right triangle with a = √(c² − b²): why you subtract, five worked examples, a mental-math shortcut and impossible inputs.
- LearnSquare Roots and Radicals for the Pythagorean TheoremThe square-root toolkit for the Pythagorean theorem: perfect squares 1–30, estimating roots, simplest radical form step by step, exact vs decimal answers.
- LearnConverse of the Pythagorean TheoremThe converse of the Pythagorean theorem explained: the statement, a clear proof, how to test any three side lengths, and many yes-or-no worked examples.
- 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.
Milestone: you can find any missing side of a right triangle, give the answer exactly and as a decimal, and test whether three lengths form a right triangle. Check with the practice problems or the quiz. To see why the theorem is true, try the rearrangement proof next.
Level 2: Core skills
Level 2 The theorem at work across geometry. Do the distance formula before the equation of a circle, and the height of a triangle before the 30-60-90 triangle, which comes from half an equilateral triangle.
| Lesson | Key formula or idea |
|---|---|
| Acute, right or obtuse? | c2 < a2 + b2 acute, = right, > obtuse |
| Distance formula | d = √(x2 − x1)2 + (y2 − y1)2 |
| In three dimensions | d = √ℓ2 + w2 + h2 |
| Height of a triangle | h = √s2 − (b/2)2 |
| Equation of a circle | x2 + y2 = r2 |
| 45-45-90 triangle | Sides in the ratio 1 : 1 : √2 |
| 30-60-90 triangle | Sides in the ratio 1 : √3 : 2 |
- LearnThe Pythagorean Inequality: Acute, Right or Obtuse?Classify any triangle as acute, right or obtuse from its side lengths with the Pythagorean inequality: compare c² with a² + b². Proof, animation and examples.
- LearnThe Distance FormulaThe distance formula explained: derive it from the Pythagorean theorem, use it with negative coordinates, avoid sign mistakes and extend it to 3D.
- LearnThe Pythagorean Theorem in 3DThe Pythagorean theorem in 3D: find the space diagonal of a box or cube, d = √(l² + w² + h²), with an animated two-step proof, fit problems and 3D distance.
- LearnFind the Height of a Triangle with the Pythagorean TheoremFind the height of an isosceles or equilateral triangle with the Pythagorean theorem: h = √(s² − (b/2)²), h = (√3/2)s, area formulas and slant heights.
- LearnEquation of a Circle and the Pythagorean TheoremWhy the equation of a circle, x² + y² = r², is the Pythagorean theorem, plus the center-radius form, worked examples and the inside/outside test.
- LearnThe 45-45-90 TriangleThe 45-45-90 triangle rules: why the sides are in the ratio 1 : 1 : √2, how to find a leg from the hypotenuse, the diagonal of a square, and worked examples.
- LearnThe 30-60-90 TriangleThe 30-60-90 triangle ratio 1 : √3 : 2 derived from half an equilateral triangle, rules for each given side, worked examples and exact values for 30° and 60°.
Milestone: you can find distances on a grid and in a box, classify a triangle from its sides, and use the special-triangle ratios without a calculator. The distance formula worksheet and 3D worksheet are good checks.
Level 3: Trigonometry links
Level 3 Where the theorem meets sine and cosine; you should already know SOHCAHTOA. New to trigonometry? Begin with "Pythagoras or trig?".
| Lesson | Key formula or idea |
|---|---|
| Law of cosines | c2 = a2 + b2 − 2abcosC |
| Pythagorean identity | sin2θ + cos2θ = 1 |
| Pythagoras or trig? | Two sides known: theorem. A side and an angle: trig. |
| Altitude and geometric mean | h2 = pq, a2 = pc, b2 = qc |
- LearnThe Law of CosinesThe law of cosines c² = a² + b² − 2ab·cos C explained: how it extends the Pythagorean theorem, a proof by dropping an altitude, and SAS and SSS examples.
- LearnThe Pythagorean IdentityThe Pythagorean identity sin²θ + cos²θ = 1 derived from the unit circle, the related tan/sec and cot/csc identities, and how to find cos from sin by quadrant.
- LearnPythagorean Theorem or Trigonometry: Which Should You Use?When to use the Pythagorean theorem vs trigonometry (SOH CAH TOA): a decision flowchart, side-by-side worked examples and multi-step problems that need both.
- LearnThe Geometric Mean TheoremThe geometric mean theorem: the altitude on the hypotenuse makes three similar triangles, so h² = pq, a² = pc and b² = qc. Proof, examples and constructions.
Milestone: you can solve any triangle, right-angled or not, and explain why the law of cosines turns into the Pythagorean theorem when the angle is 90°.
Level 4: Going further
Level 4 Generalizations and beautiful constructions, in any order. The spiral, √2 and Pythagoras tree pages need only Level 1; higher dimensions and curved surfaces are best after Level 3.
| Lesson | Key formula or idea |
|---|---|
| Beyond squares | Any similar shapes on the sides: area on a + area on b = area on c |
| On curved surfaces | On a sphere: cosc over R = cosa over Rcosb over R |
| In n dimensions | ‖v‖2 = v12 + v22 + ⋯ + vn2 |
| De Gua's theorem | The squared areas of the three right-angled faces add up to the squared area of the fourth face |
| Spiral of Theodorus | (√n)2 + 12 = n + 1 |
| Why √2 is irrational | 12 + 12 = 2, and √2 is not a fraction |
| Pythagoras tree | A fractal built by repeating squares on a right triangle |
- LearnThe Pythagorean Theorem with Semicircles and Other ShapesEuclid VI.31 extends the Pythagorean theorem from squares to semicircles, equilateral triangles and any similar shapes. See why it works, with examples.
- LearnThe Pythagorean Theorem on a Sphere and in Non-Euclidean GeometryDoes a² + b² = c² hold on a sphere? The spherical and hyperbolic Pythagorean theorems, why they shrink to the usual one for small triangles, and Earth examples.
- LearnThe Pythagorean Theorem in n DimensionsThe Pythagorean theorem in n dimensions: the Euclidean norm, orthogonal vectors and the dot product, inner-product spaces, Parseval's identity and data science.
- LearnDe Gua's TheoremDe Gua's theorem says the squared area of the slanted face of a right-corner tetrahedron equals the sum of the other three. Proofs, examples and history.
- LearnThe Spiral of TheodorusThe spiral of Theodorus chains right triangles with a leg of 1 to build √2, √3, √4 … √17. How to draw it, why it stops at √17, its history and a class activity.
- LearnWhy the Square Root of 2 Is IrrationalWhy the square root of 2 is irrational: the classic even-odd proof, Tennenbaum's picture proof, the Hippasus legend, the YBC 7289 tablet and √2's continued fraction.
- LearnThe Pythagoras Tree FractalThe Pythagoras tree fractal (Albert E. Bosman, 1942): how it is built from squares, why every level has the same area, lopsided trees and code to draw one.
Reference pages
Useful at every level: the cheat sheet (every formula on one page), the calculators (answers with full working) and the proofs (why it is true).
- 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.
- LearnThe Pythagorean Theorem for KidsThe Pythagorean theorem explained simply for kids aged 9–13: counting tiles, a shortcut across the park, screen and pizza-box examples, and five easy questions.
Teachers will find lesson plans and activities in the teaching guide; the story behind the theorem is in the history section.
Suggested study order
- Test tomorrow: what the theorem is, how to use it, common mistakes, the cheat sheet, then practice problems.
- Grade 8 unit: Level 1 in order, the distance formula and 3D lessons, and one proof. This matches the US Common Core: explain a proof, find unknown sides in 2D and 3D, and find distances between points.
- High school geometry: the self-check, then Level 2 and the word problems.
- Precalculus: Level 3, then any Level 4 pages.
Frequently asked questions
What do I need to know before learning the Pythagorean theorem?
How to square a number, what a square root is, what a right angle is, and how to solve a simple equation such as x + 16 = 25. The prerequisites section above has a quick refresher on each.
How long does it take to learn the Pythagorean theorem?
The main idea fits in one lesson. Fluency, meaning you can spot the hypotenuse in any diagram, choose between adding and subtracting, and handle radicals and word problems, takes several practice sessions. Level 1 of this course is built for exactly that.
What grade do you learn the Pythagorean theorem in?
In the United States it is part of the grade 8 Common Core standards, and it comes back in high school geometry, trigonometry and precalculus.
Do I need trigonometry to use the Pythagorean theorem?
No. The theorem only needs squares and square roots, and it is usually taught before trigonometry. Trigonometry becomes necessary when you know an angle rather than two sides; Level 3 shows how the two fit together.
Keep exploring
- PracticePythagorean Theorem PracticePythagorean theorem practice in five levels, from finding the hypotenuse to challenge problems, with warm-ups, a one-week plan and every worksheet and quiz.
- ProofsProofs of the Pythagorean TheoremProofs of the Pythagorean theorem: an animated proof, how a proof differs from checking examples, all 12 proofs on this site by type, and where to start.
- CalculatorsPythagorean Theorem CalculatorsEvery free Pythagorean theorem and right triangle calculator on the site, with a guide to choosing the right one and tips on exact answers, rounding and units.
- PracticePythagorean Theorem Cheat SheetPrintable Pythagorean theorem cheat sheet: the formula and its rearrangements, the converse, special right triangles, triples, distance formulas and squares.
- PracticePythagorean Theorem QuizTake a free Pythagorean theorem quiz: 10 questions on the hypotenuse, missing sides, triples, the converse and distance, with instant worked solutions.