Pythagorean Theorem — Mathematics Knowledge System

Overview

The Pythagorean theorem relates the three sides of a right triangle: the square of the hypotenuse equals the sum of the squares of the other two sides. It is one of the oldest and most-proved theorems in mathematics, with hundreds of distinct known proofs, and underlies the definition of distance in geometry, trigonometry, and beyond.

Intuition

For any right triangle, build a square directly on each of its three sides. The theorem says the area of the square on the longest side (hypotenuse) exactly equals the combined area of the two squares on the other sides — no matter how you stretch or shrink the right angle's legs, that area relationship never breaks. It's a statement about area conservation, not just a formula to memorize.

Formal Definition

Definition

For a right triangle with legs a, b and hypotenuse c:

a^2 + b^2 = c^2

Pythagorean theorem

Notation

Notation	Meaning
$a, b$	The two legs (sides adjacent to the right angle)
$c$	The hypotenuse (side opposite the right angle, always the longest)

Proofs

Proof by rearrangement

$\text{Arrange 4 copies of the right triangle inside a square of side } (a+b)$ (Construction)
$\text{The 4 triangles leave an inner square of side } c \text{ uncovered}$ (By construction, the inner boundary forms a square)
$(a+b)^2 = 4 \cdot \tfrac{1}{2}ab + c^2$ (Total area = 4 triangle areas + inner square area)
$a^2 + 2ab + b^2 = 2ab + c^2 \implies a^2+b^2=c^2$ (Expand and cancel the 2ab terms)

Properties

Pythagorean triple

\text{Integer solutions to } a^2+b^2=c^2

Example: (3,4,5), (5,12,13), (8,15,17)

Converse also holds

\text{If } a^2+b^2=c^2 \text{ for a triangle's sides, the triangle must have a right angle.}

Applications

Distance calculations in 2D/3D CAD, construction (squaring corners with the 3-4-5 rule), and structural load analysis all use the Pythagorean relationship.

Interactive Geometry

Worked Examples

Apply the theorem directly.
$c = \sqrt{3^2+4^2} = \sqrt{25} = 5$

Answer: 5

Practice Problems

Difficulty 2/10

A ladder 13 ft long leans against a wall, its base 5 ft from the wall. How high up the wall does it reach?

Common Mistakes

Common Mistake

Applying a² + b² = c² to a triangle that isn't a right triangle.

The theorem ONLY applies to right triangles. For other triangles, use the Law of Cosines, which generalizes it: c² = a² + b² - 2ab·cos(C).

Common Mistake

Mislabeling which side is the hypotenuse.

The hypotenuse is always the side OPPOSITE the right angle, and always the longest side — squaring the wrong side gives a wrong answer even with correct arithmetic.

Quiz

In a right triangle with legs 6 and 8, what is the hypotenuse?

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Flashcards

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Historical Background

Babylonian mathematicians knew and used Pythagorean triples (like 3-4-5) at least a thousand years before Pythagoras, as shown by the clay tablet Plimpton 322 (c. 1800 BCE). The theorem is attributed to Pythagoras (c. 570-495 BCE) because ancient Greek tradition credits his school with the first general proof, though no writing of his survives directly. Euclid gave a rigorous proof in his Elements (c. 300 BCE, Book I, Proposition 47), and the theorem has since accumulated hundreds of distinct proofs, including one by future U.S. President James Garfield in 1876.

c. 1800 BCE
Babylonian tablet Plimpton 322 lists Pythagorean triples
c. 500 BCE
Pythagoras's school credited with the first general proof
Pythagoras
c. 300 BCE
Euclid publishes a rigorous geometric proof in the Elements
Euclid

Summary

For a right triangle: a² + b² = c², where c is the hypotenuse.
Known to Babylonians ~1000 years before Pythagoras; proved rigorously by Euclid.
The converse also holds: a²+b²=c² for a triangle's sides implies a right angle.
Generalizes to the Law of Cosines for non-right triangles.
Foundation of Euclidean distance used throughout geometry, graphics, and ML.

References

WebsiteWikipedia — Pythagorean theorem