This entry was posted on December 26, 2021 by Anne Helmenstine (updated on January 3, 2022)

In mathematic, the **Pythagorean theorem** states that **the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two sides**. Another way of stating the theorem is that the sum of the areas of the squares formed by the sides of a right triangle equals the area of the square whose side is the hypotenuse. The theorem is a key relation in Euclidean geometry. It is named for the Greek philosopher Pythagorus.

Remember: The Pythagorean theorem only applies to right triangles!

### Pythagorean Theorem Formula

The formula for the Pythagorean theorem describes the relationship between the sides *a* and *b* of a right triangle to its hypotenuse, *c*. A right triangle is one containing a 90° or right angle. The hypotenuse is the side of the triangle opposite from the right angle (which is the largest angle in a right triangle).

*a*^{2} + *b*^{2} = *c*^{2}

### Solving for a, b, and c

Rearranging the equation gives the formulas solving for a, b, and c:

- a = (c
^{2}– b^{2})^{½} - b = (c
^{2}– a^{2})^{½} - c = (a
^{2}+ b^{2})^{½}

### How to Solve the Pythagorean Theorem – Example Problems

For example, find the hypotenuse of a right triangle with side that have lengths of 5 and 12.

Start with the formula for the Pythagorean theorem and plug in the numbers for the sides *a* and *b* to solve for *c*.

*a*^{2} + *b*^{2} = *c*^{2}

5^{2} + 12^{2} = *c*^{2}*c*^{2} = 5^{2} + 12^{2} = 25 + 144 = 169

c2 = 169

c = √169 or 169** ^{½}** = 13

For example, solve for side b of a triangle where *a* is 9 and the hypotenuse *c* is 15.

*a*^{2} + *b*^{2} = *c*^{2}

9^{2} + b^{2} = 15^{2}

b^{2} = 15^{2} – 9^{2} = 225 – 81 = 144

b = √144 = 12

Now, let’s combine a bit of algebra with the geometry. Solve for x where the sides of a right triangle are 5x and 4x +5 and the hypotenuse has a length of 8x -3.

*a*^{2} + *b*^{2} = *c*^{2}

(5x)^{2} + (4x +5)^{2} = (8x-3)^{2}

The (4x + 5)^{2} and (8x -3)2 terms are the squares of binomial expressions. So, expanding the equation gives the following:

25x^{2} + (4x +5)(4x +5) = (8x -3)(8x -3)

25x^{2} = 16×2 + 20x +20x + 25 = 64x – 24x – 24x + 9

Combine like terms:

41x^{2} + 40x + 25 = 64x^{2} – 48x + 9

Rewrite the equation and solve for x.

0 = 23x^{2} – 88x – 16

Apply the quadratic equation and solve for x:

x = [-b **±** √(b^{2} -4ac)]/2a

x = [-(-88) **±** √[-88^{2} – 4(23)(-16)] / 2(23) = [88 **±** √(7744 + 1472)] / 46 = (88 **±** 96) / 46

So, there are two answers:

x = (88 + 96)/46 = 4 and (88 – 96).46 = -4/23

A triangle does not have a negative length for its side, so x is 4.

Plugging in”4″ in place of x, the sides of the right triangle are 20, 21, and 29.

### Pythagorean Triples

Pythagorean triples are integers a, b, and c, that represent the sides of a right triangle and satisfy the Pythagorean theorem. Here is the list of Pythagorean triples for integers with values less than 100:

<!-MONUMETRIC Repeatable 2 D:300x250 T:300x250 M:300x250,320x50 START->

<!-MONUMETRIC Repeatable 2 D:300x250 T:300x250 M:300x250,320x50 ENDS->

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

### Proof of the Pythagorean Theorem

There are more proofs for the Pythagorean theorem than for any other theorem in geometry! At least 370 proofs are known. Some of these proofs use the parallel postulate. Some rely on the complementarity of acute angles in a right triangle. Proofs using shearing use the properties of parallelograms.

### History – Did Pythagoras Discover the Pythagorean Theorem?

While the Pythagorean theorem takes its name from Pythagorus, he did not discover it. Exactly who gets the credit or whether many different places made the discovery independently is a matter of debate. The Mesopotamians made calculations using the formula as early as 2000 BC, which was over a thousand years before Pythagorus. A papyrus from the Egyptian Middle Kingdom, dating between 2000 and 1786BC, references a math problem describing Pythagorean triples. The *Baudhayana Shulba Sutra* from India (dating between the 8th and 5th century BC) lists both Pythagorean triples and the Pythagorean theorem. The “Gougu theorem” from China offers a proof for the Pythagorean theorem, which came into use long before its oldest surviving description from the 1st century BC.

Pythagorus of Samos lived between 570 and 495 BC. While he was not the original person who formulated the Pythagorean theorem, he (or his students) may have introduced its proof to ancient Greece. In any case, his philosophical treatment of math left a lasting impression on the world.

### References

- Bell, John L. (1999).
*The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development*. Kluwer. ISBN 0-7923-5972-0. - Heath, Sir Thomas (1921). “The ‘Theorem of Pythagoras‘”.
*A History of Greek Mathematics*(2 Vols.) (Dover Publications, Inc. (1981) ed.). Oxford: Clarendon Press. ISBN 0-486-24073-8. - Maor, Eli (2007).
*The Pythagorean Theorem: A 4,000-Year History*. Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-12526-8. - Swetz, Frank; Kao, T. I. (1977).
*Was Pythagoras Chinese?: An Examination of Right Triangle Theory in Ancient China*. Pennsylvania State University Press. ISBN 0-271-01238-2.