What is the Pythagorean Theorem?
The Pythagorean theorem relates the three sides of a right triangle: the sum of the squares of the two legs equals the square of the hypotenuse. It's one of the oldest and most-used results in geometry, essential for distance, construction and navigation problems.
The Pythagorean theorem states 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).
- 1↓Identify legs a and bThe two sides that meet at the right angle.
- 2↓Square each legCompute a² and b².
- 3↓Add the squaresSum a² + b².
- 4Take the square rootc = √(a² + b²).
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Step-by-step worked examples
A right triangle has legs 3 and 4. Find the hypotenuse.
a² + b² = c² 3² + 4² = 9 + 16 = 25 c = √25 = 5
A ladder leans against a wall, reaching 8 m up while its base is 6 m from the wall. How long is the ladder?
a = 6, b = 8 c² = 6² + 8² = 36 + 64 = 100 c = √100 = 10 m
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
a² + b² = c² 5² + b² = 13² 25 + b² = 169 b² = 144 b = 12
Flashcards
Quick quiz
Q1.Legs 6 and 8 — what is the hypotenuse?
Q2.The Pythagorean theorem applies to:
Q3.Hypotenuse = 13, leg = 12. Other leg?
Q4.Which side is always the longest in a right triangle?
The full card deck, worked steps and AI-tutor support for “What is the Pythagorean Theorem?” are in Notek — study by hand before your exam.
Common mistakes
Using the theorem on a non-right triangle. — Correct: It only applies when one angle is exactly 90°; otherwise use the Law of Cosines.
Adding the sides instead of squaring them. — Correct: You must square each leg first: a² + b², not a + b.
Mislabeling the hypotenuse as a leg. — Correct: The hypotenuse is always the side opposite the right angle, and always the longest.
Forgetting to take the square root at the end. — Correct: c² gives the square of the hypotenuse — take √ to get c itself.
FAQ
What is the Pythagorean theorem?
A formula relating a right triangle's sides: a² + b² = c², where c is the hypotenuse.
What is the Pythagorean theorem formula?
a² + b² = c². Square both legs, add them, then take the square root for the hypotenuse.
How do you calculate the Pythagorean theorem?
Square the two legs, add the results, then take the square root of the sum to get the hypotenuse.
What are examples of the Pythagorean theorem?
Common integer examples (Pythagorean triples) include 3-4-5, 6-8-10, and 5-12-13.