How The Pythagorean Theorem Helps Solve a Right Triangle

In its simplest form, the Pythagorean theorem states that in a hypothetical right triangle abc: a² + b² = c².

The value of c² is equal to the sum of the squares, where hypotenuse c is the longest side of a right triangle. It's also always the side opposite the right angle.

Using this formula, we can always find the length of the hypotenuse if the other two sides are known values. After adding the numbers, we will need to apply a square root operation to arrive at the value of c.

Going back to triangle abc, what do we do if one of the known sides is the hypotenuse? We can reverse the Pythagorean theorem formula and turn it into a subtraction problem, then apply a square root just as before.

If a triangle contains two unknown sides, then more complex trigonometric formulas and algebraic proofs will have to be applied in order to find them. This same mathematical theorem can also be applied to physics problems like triangular force vectors.

What Is a Right Triangle?

A right angled triangle has exactly one of its angle values equal to 90 degrees, which is where the Pythagorean theorem formula can be applied. The side opposite the right angle is known as the hypotenuse and will always be the longest side of the right angled triangle.

Triangles without a right angle, like a scalene or isosceles triangle, cannot be solved using the Pythagorean theorem. They must be broken up into smaller shapes or have more complex formulas applied.

Like all triangles, the angle values of a right angled triangle add up to a sum of 180 degrees. This also means that the two non-right angles of the triangle must add up to 90 degrees.

Now that we know a bit about solving right angled triangle abc, let's replace our variables with real numbers and run through the formula again. The side lengths we know are 16 and 20, and our hypotenuse is the unknown side.

Based on these calculations, we now know that the hypotenuse of the triangle equals 25.61.

Another useful idea related to Pythagoras theorem proof is the concept of a Pythagorean triple. These are essentially forms of right triangles which have sides that are all equal to whole numbers.

The most common form of a Pythagorean triple you are likely to see math education is known as the (3, 4, 5) triangle. If two sides of a right triangle equal 3 and 4, then the hypotenuse will always be 5.

Shown using the Pythagorean theorem formula:

Learning to spot Pythagorean triples by eye can help you easily solve them without resorting to the Pythagorean theorem formula every time. There are theoretically infinite Pythagorean triples out there, but some other common ones include:

Similar triangles to a Pythagorean triple will themselves be triples. So we can multiply all the values of our previous example by 2 to get a triangle of (6, 8, 10).

Multiplying any Pythagorean triple by any positive integer (as in, applying the same multiplier to all sides of the same triangle) will give you similar results.

The Pythagorean equation is most often attributed to Pythagoras of Somos, but we now know that many ancient civilizations like those in Egypt, India and China had discovered the mathematical relationship independently.

That said, the man whom this math trick is named for is nearly as fascinating. Pythagoras, an ancient Greek thinker who was born on the island of Samos and lived from 570 to 490 B.C.E, was kind of a trippy character — equal parts philosopher, mathematician and mystical cult leader.

In his lifetime, Pythagoras wasn't known as much for solving for the length of the hypotenuse as he was for his belief in reincarnation and adherence to an ascetic lifestyle that emphasized a strict vegetarian diet, adherence to religious rituals and plenty of self-discipline that he taught to his followers.

Pythagoras biographer Christoph Riedweg describes him as a tall, handsome and charismatic figure, whose aura was enhanced by his eccentric attire — a white robe, trousers and a golden wreath on his head. Odd rumors swirled around him — that he could perform miracles, that he had a golden artificial leg concealed beneath his clothes and that he possessed the power to be in two places at one time.

Pythagoras founded a school near what is now the port city of Crotone in southern Italy, which was named the Semicircle of Pythagoras. Followers, who were sworn to a code of secrecy, learned to contemplate numbers in a fashion similar to the Jewish mysticism of Kaballah. In Pythagoras' philosophy, each number had a divine meaning, and their combination revealed a greater truth.

With a hyperbolic reputation like that, it's little wonder that Pythagoras was credited with devising one of the most famous theorems of all time, even though he wasn't actually the first to come up with the concept. Chinese and Babylonian mathematicians beat him to it by a millennium.

"What we have is evidence they knew the Pythagorean relationship through specific examples," writes G. Donald Allen, a math professor and director of the Center for Technology-Mediated Instruction in Mathematics at Texas A&M University, in an email. "An entire Babylonian tablet was found that shows various triples of numbers that meet the condition: a² + b² = c²."

The Pythagorean theorem isn't just an intriguing mathematical exercise. It's utilized in a wide range of fields, from construction and manufacturing to navigation.

As Allen explains, one of the classic uses of the Pythagorean theorem is in laying the foundations of buildings. "You see, to make a rectangular foundation for, say, a temple, you need to make right angles. But how can you do that? By eyeballing it? This wouldn't work for a large structure. But, when you have the length and width, you can use the Pythagorean theorem to make a precise right angle to any precision."

Beyond that, "This theorem and those related to it have given us our entire system of measurement," Allen says. "It allows pilots to navigate in windy skies, and ships to set their course. All GPS measurements are possible because of this theorem."

In navigation, the Pythagorean theorem provides a ship's navigator with a way of calculating the distance to a point in the ocean that's, say, 300 miles north and 400 miles west (480 kilometers north and 640 kilometers west). It's also useful to cartographers, who use it to calculate the steepness of hills and mountains.

"This theorem is important in all of geometry, including solid geometry," Allen continues. "It is also foundational in other branches of mathematics, much of physics, geology, all of mechanical and aeronautical engineering. Carpenters use it and so do machinists. When you have angles, and you need measurements, you need this theorem."