the Pythagorean theorem
/ pih-THAG-or-as (for Pythagoras) /
The Pythagorean theorem is the most famous result in elementary geometry: in any right triangle, the square built on the longest side equals the sum of the squares built on the other two. If the two short sides (the legs) have lengths a and b, and the longest side (the hypotenuse, opposite the right angle) has length c, then a^2 + b^2 = c^2. It connects the three sides of a right triangle by a single, exact equation.
Read it as areas: literally, the area of a square drawn on the hypotenuse equals the combined area of the squares drawn on the two legs. From that one relation you can find any side when the other two are known — for instance with legs 3 and 4, c^2 = 9 + 16 = 25, so c = 5. There are hundreds of proofs. A rearrangement proof fits four copies of the triangle inside a big square two ways to compare areas; a similarity proof drops the altitude to the hypotenuse and adds the two geometric-mean relations; Euclid's own proof (Elements I.47, the 'windmill' or 'bride's chair') shows each leg-square equals a rectangle on the hypotenuse by congruent triangles.
Its reach is enormous: it gives the distance between two points in the coordinate plane (the distance formula is the theorem in disguise), it underlies trigonometry's Pythagorean identity, and it generalises into three dimensions and beyond. One honest caution: the theorem is an if-then statement that requires a right angle. It does not hold in a triangle that is not right-angled, and on a curved surface like a sphere it fails entirely — it is a theorem of flat (Euclidean) geometry. The credit to Pythagoras is traditional; Babylonian tablets show the relation was known and used a thousand years before him.
A ladder 13 m long leans against a wall with its foot 5 m from the base. How high does it reach? With c = 13 and a = 5, b^2 = 13^2 - 5^2 = 169 - 25 = 144, so b = 12 m.
a^2 + b^2 = c^2 lets you recover any side of a right triangle from the other two.
The theorem applies only to right triangles, and c must be the hypotenuse (the side opposite the right angle, always the longest). A frequent error is to write a^2 + b^2 = c^2 with c as a leg; if you put the wrong side as c, every answer is wrong.