Pythagorean Theorem

The Pythagorean Theorem is a mathematical formula used for finding missing lengths of right triangles. It is named after the Greek mathematician Pythagoras, who by tradition is often credited with its discovery and proof, though many speculate that knowledge of the Pythagorean Theorem predates him.

According to the Pythagorean Theorem,

"In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs."

In other words,
 * $$a^2 + b^2 = c^2$$.

This is read as, "a squared plus b squared equals c squared."

How It Works
Let's say you were to find the length of side a, where the length of b is 4 and the length of c is 5.


 * a2 + b2 = c2
 * a2 + 42 = 52
 * a2 + 16 = 25
 * a2 = 9

Since the square root of 9 is 3, then a = 3.

Converse
If a2 + b2 = c2, then the angle between a and b is 90°.

Pythagorean Triples
Whole number solutions to the Pythagorean equation are known as Pythagorean triples. The Ancient Greeks proved that there are infinitely many of these.

Suppose we have two numbers m and n. Assume m > n.
 * a = m2 − n2
 * b = 2mn
 * c = m2 + n2

Do note that a is a difference of two squares, while c is a sum of two squares. You can read up on properties of sums and differences of two squares here. Moreover, because b is defined as a multiple of two, this means that, in a Pythagorean triple, at least one of these numbers must be even.

We can verify these values for a, b, and c using the FOIL method.
 * a2 = (m2 − n2)2 = (m2 − n2)(m2 − n2) = m4 − 2m2n2 + n4
 * b2 = (2mn)2 = 4m2n2
 * c2 = (m2 + n2)2 = (m2 + n2)(m2 + n2) = m4 + 2m2n2 + n4

Now let's add a2 and b2.
 * a2 + b2 = m4 − 2m2n2 + n4 + 4m2n2
 * = m4 + 2m2n2 + n4 = c2. Q.E.D.

Here are some examples of Pythagorean triples:
 * 32 + 42 = 52
 * 52 + 122 = 132
 * 72 + 242 = 252
 * 92 + 402 = 412
 * 82 + 152 = 172

Shortcuts
There are special types of right triangles in which the Pythagorean Theorem can be used to find missing lengths easily.

45-45-90 Triangles
Full article: Triangles A 45-45-90 triangle is a right triangle with the angles measuring at 45° and 90° only.

Here is the formula for finding the missing lengths of 45-45-90 triangles.

Since $$a = b$$, and $$x = a$$ or $$b$$, then the formula becomes


 * $$x^2 + x^2 = c^2$$, or


 * $$2x^2 = c^2$$.

The shortcut for finding the length of c:


 * $$c = x\sqrt{2}\,$$.

For example, say that the value of x is 7. That would mean that the value of c is $$7\sqrt{2}\,$$.

To find x,


 * $$x = \frac{c}{\sqrt{2}\,}\,$$, or


 * $$x = \frac{c\sqrt{2}\,}{2}\,$$.

Did You Know? You can find two 45-45-90 triangles when you bisect a square.

30-60-90 Triangles
Full article: Triangles A 30-60-90 triangle is a right triangle with the angles measuring at 30°, 60°, and 90°. Two of these can be found by bisecting an equilateral triangle.

Since the sides of an equilateral triangle are the same length, then


 * $$c = 2a$$.

Therefore, the formula becomes


 * $$a^2 + b^2 = (2a)^2$$, or


 * $$a^2 + b^2 = 4a^2$$.

To find the length of b,


 * $$b = a\sqrt{3}\,$$.

To find a,


 * $$a = \frac{b}{\sqrt{3}\,}\,$$, or


 * $$a = \frac{b\sqrt{3}\,}{3}\,$$.

Acute Triangles
If a triangle is acute, then $$a^2 + b^2 < c^2$$.

Obtuse Triangles
If a triangle is obtuse, then $$a^2 + b^2 > c^2$$.