Pythagoras' Theorem explains a key relationship in right-angled triangles, stating that the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides.

This timeless math principle, often written as a2+b2=c2a^2+b^2=c^2a2+b2=c2, applies only to right triangles where one angle is exactly 90 degrees.

Core Formula

In a right triangle:

  • Let aaa and bbb be the legs (sides forming the right angle).
  • Let ccc be the hypotenuse (opposite the right angle).

a2+b2=c2a^2+b^2=c^2a2+b2=c2

Example : For sides 3, 4, and hypotenuse 5: 32+42=9+16=25=523^2+4^2=9+16=25=5^232+42=9+16=25=52. This "3-4-5" triple is one of many Pythagorean triples used in construction and navigation.

Quick History

Ancient Greek mathematician Pythagoras (c. 570–495 BCE) popularized it, though Babylonians knew it earlier around 1800 BCE via clay tablets. Over 300 proofs exist, from Euclid's geometry to Einstein's dissection method.

"In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides."

Real-World Uses

  • Navigation : GPS and maps calculate straight-line distances.
  • Architecture : Ensures square corners in buildings.
  • Physics : Models distances in vectors or forces.

Recent trends (as of March 2026) tie it to AI-optimized proofs or viral TikTok explainers, but the core stays unchanged.

Application| Example Calculation
---|---
Find hypotenuse (legs 6, 8)| c=62+82=36+64=10c=\sqrt{6^2+8^2}=\sqrt{36+64}=10c=62+82​=36+64​=10 7
Find leg (hypotenuse 13, leg 5)| b=132−52=169−25=12b=\sqrt{13^2-5^2}=\sqrt{169-25}=12b=132−52​=169−25​=12 10
Verify right triangle (sides 7, 24, 25)| 72+242=49+576=625=2527^2+24^2=49+576=625=25^272+242=49+576=625=252 6

Proof Sketch (Similarity Method)

Drop a perpendicular from the right angle to the hypotenuse, creating similar triangles. Ratios yield c2=a2+b2c^2=a^2+b^2c2=a2+b2.

TL;DR : a2+b2=c2a^2+b^2=c^2a2+b2=c2 for right triangles—simple, powerful, eternal.

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