What the Pythagorean theorem states
In any right triangle -- a triangle containing exactly one 90-degree angle -- the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides, called legs. Written as a formula: a^2 + b^2 = c^2, where a and b are the two legs and c is the hypotenuse. The relationship is named for the ancient Greek mathematician Pythagoras, although evidence shows the same numerical relationship was known and used, independently of any formal proof, by Babylonian and other ancient mathematicians well before Pythagoras.
The theorem applies only to right triangles on a flat (Euclidean) plane. It does not hold for triangles that lack a right angle -- those require the more general law of cosines instead -- and it does not hold exactly for triangles drawn on a curved surface, such as a sphere, where the geometry itself is different.
Why it works: a classical rearrangement proof
One of the best-known demonstrations of the theorem, often called the rearrangement proof, uses four identical copies of the same right triangle (with legs a and b and hypotenuse c) arranged inside a single large square with sides of length (a + b). Arranged one way, the four triangles surround a smaller, tilted inner square with sides of length c, so the large square's area splits into the four triangles plus that inner square: (a + b)^2 = 4 x (1/2 x a x b) + c^2, which simplifies to (a + b)^2 = 2ab + c^2.
Expanding the left side algebraically, (a + b)^2 = a^2 + 2ab + b^2. Setting the two expressions for the same total area equal to each other: a^2 + 2ab + b^2 = 2ab + c^2. Subtracting 2ab from both sides leaves a^2 + b^2 = c^2 -- the Pythagorean theorem, derived purely from comparing two ways of calculating the same square's area. This particular proof is one of hundreds that have been published over the centuries; Elisha Loomis's compendium The Pythagorean Proposition catalogs 367 distinct proofs of the theorem.
Pythagorean triples: 3-4-5 and beyond
A Pythagorean triple is a set of three positive whole numbers a, b and c that satisfy a^2 + b^2 = c^2 exactly, meaning a right triangle with those three whole-number side lengths exists with no rounding required. The smallest and most widely recognized triple is 3-4-5, since 3^2 + 4^2 = 9 + 16 = 25 = 5^2.
Any whole-number multiple of a known triple is also a valid triple -- for example, doubling 3-4-5 gives 6-8-10, and tripling it gives 9-12-15, both of which also satisfy a^2 + b^2 = c^2. Triples that are not multiples of a smaller triple, such as 3-4-5, 5-12-13 and 8-15-17 below, are called primitive Pythagorean triples.
| Leg a | Leg b | Hypotenuse c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 9 | 40 | 41 |
| 20 | 21 | 29 |
Real uses: diagonals and distances
Because a rectangle's diagonal, together with two adjacent sides, forms a right triangle, the Pythagorean theorem gives the diagonal length directly: diagonal = the square root of (length^2 + width^2). A rectangle measuring 6 by 8 units has a diagonal of the square root of (6^2 + 8^2) = the square root of (36 + 64) = the square root of 100 = exactly 10 units -- the 6-8-10 triple, a multiple of 3-4-5.
The same principle extended into three dimensions gives the space diagonal of a rectangular box, found by applying the theorem twice: first to find the diagonal of the base rectangle, then combining that diagonal with the box's height. A box measuring 12 by 9 by 8 units has a space diagonal of the square root of (12^2 + 9^2 + 8^2) = the square root of (144 + 81 + 64) = the square root of 289 = exactly 17 units. The coordinate distance formula used to find the straight-line distance between two points, d = the square root of ((x2-x1)^2 + (y2-y1)^2), is the same theorem applied directly to the horizontal and vertical differences between two points: for the points (1,2) and (4,6), the distance is the square root of (3^2 + 4^2) = the square root of 25 = 5.
The converse: using 3-4-5 to check a right angle
The converse of the Pythagorean theorem is also true and is frequently used in practice: if three measured side lengths satisfy a^2 + b^2 = c^2, then the triangle they form must contain a right angle opposite the longest side, even if that angle was never measured directly. This converse is the basis of the traditional 3-4-5 method used in construction and carpentry to check or set a square (90-degree) corner: measuring 3 units along one edge, 4 units along the perpendicular edge, and confirming the diagonal between those two marked points is exactly 5 units confirms the corner is a true right angle.
This works because the converse guarantees a unique outcome: three positive lengths satisfying a^2 + b^2 = c^2 can only form a right triangle, never an acute or obtuse one, with the right angle located opposite the longest side, c. Any multiple of the 3-4-5 triple -- such as 6-8-10 or 9-12-15 -- works identically for checking larger corners.
Domande frequenti
What is the Pythagorean theorem formula?
a^2 + b^2 = c^2, where a and b are the two legs of a right triangle and c is the hypotenuse, the side opposite the right angle. It applies only to right triangles on a flat plane.
How do you prove the Pythagorean theorem?
One classical method, the rearrangement proof, arranges four identical right triangles inside a square with sides (a + b) so that the total area can be calculated two different ways -- once as (a + b)^2, and once as the four triangles plus a tilted inner square of area c^2. Setting these two area expressions equal and simplifying algebraically produces a^2 + b^2 = c^2. Hundreds of alternative proofs have also been published, including Elisha Loomis's compendium documenting 367 of them.
What is a Pythagorean triple, and what is the smallest one?
A Pythagorean triple is a set of three positive whole numbers a, b and c satisfying a^2 + b^2 = c^2 exactly. The smallest and most common triple is 3-4-5, since 3^2 + 4^2 = 9 + 16 = 25 = 5^2. Other common triples include 5-12-13, 8-15-17, 7-24-25 and 9-40-41.
How is the Pythagorean theorem used to find the diagonal of a rectangle?
A rectangle's diagonal forms the hypotenuse of a right triangle with the rectangle's length and width as the two legs, so diagonal = the square root of (length^2 + width^2). A 6-by-8 rectangle has a diagonal of the square root of (36 + 64) = the square root of 100 = exactly 10 units.
Does the Pythagorean theorem work on curved surfaces?
No. The theorem holds exactly only for right triangles drawn on a flat (Euclidean) plane. On a curved surface, such as the surface of a sphere, the relationships between angles and side lengths are governed by a different, non-Euclidean geometry, and a^2 + b^2 = c^2 no longer holds exactly for a triangle with a 90-degree angle.
How do builders use the 3-4-5 rule to check a right angle?
By the converse of the Pythagorean theorem, if three measured lengths satisfy a^2 + b^2 = c^2, the angle opposite the longest length must be exactly 90 degrees. Marking 3 units along one edge and 4 units along a perpendicular edge, then confirming the distance between those two points is exactly 5 units, verifies a true right-angle corner -- a method commonly used in construction and carpentry.
Fonti
- Weisstein, Eric W. "Pythagorean Theorem" and "Pythagorean Triple." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- Loomis ES. The Pythagorean Proposition. National Council of Teachers of Mathematics (NCTM), 1968 (a compendium of 367 proofs).
- Euclid. Elements, Book I, Proposition 47 (the classical geometric proof of the theorem).