How to Make a Right Angle

Couple things – honesty and accepting responsibility may be difficult to do in the moment but will ultimately lead to happiness. And – oddly enough, the pythagorean theorem. I smile every year when I use it to make sure my backyard hockey rink is square

TW (former student)

My response:

Amazing isn’t it? Every time I build something and need that perfect square I measure  3“ on one side, 4” on the other, and it had better be 5” from mark to mark. I get such joy doing that! Oh, and being honest and responsible are pretty important, too!

More about Pythagoras:

Pythagoras was an Ancient Greek philosopher/mathematician/cult leader. He lived in the late 6th century BCE. While born in Samos, Pythagoras settled in Croton in Southern Italy. It is thought that he traveled extensively. His main ideas and teachings may have been shaped by things he learned when traveling in Egypt. There is a lot of speculation about him and the people that followed his teachings (Pythagoreans). This is due to the fact that the majority of stories about him come from Diogenes and Iamblichus who wrote about him hundreds of years later. 

Pythagoreans had some rules that didn’t really help with people getting to know much about their leader. One was a vow of silence so as not to share their ideas with the uninitiated. That makes it a little difficult to get the word out about his teachings. Other ritual conduct included vegetarianism- but no eating of beans. Pacifism, but ironically, not following many of their rules resulted in execution. Pythagoreans were to live simply, pursuing both spiritual and mathematical knowledge. The thought being that these were the same. Belief that the soul was separate from the body and would be reincarnated. That is probably why vegetarianism was so important. You didn’t want to accidentally eat your old friend Rob who was reincarnated as a deer!

But what about the theorem?! We all learned that for any right triangle, the sum of the squares of the legs of the triangle are equal to the square of its hypotenuse. Or, 

a² + b² = c²

The biggest qualifier is that it has to be a right triangle, which means the largest angle has to be 90°. It won’t work with acute, obtuse, or even equilateral triangles*. There are special right triangles like the 3, 4, 5 mentioned at the beginning. There is also the 30-60-90 and the right isosceles. 

So where do “normal people” use this theorem? It’s usually used in measuring. 

• Going 6 blocks North and 8 blocks East? You are actually 10 blocks in a straight line from start to finish. 
• As mentioned before, the 3-4-5 right triangle is an efficient and reliable way to check if your angles are square. It is used all the time when framing walls, laying tiles, and setting foundations.

Have you ever noticed that your 60 inch television isn’t 60” in width or height? The 60” is the diagonal measurement. That TV is actually 52” wide and 29” tall. 

There are many ways to show the Pythagorean Theorem. Mathematically is fun, but I prefer a physical model that I first saw at the Museum of Science- Boston. A fluid would completely fill the square made from the hypotenuse of a right triangle. When you turn the triangle upside down the fluid fills the other two squares made from the legs of the triangle. A simple and elegant way to show that a² + b² = c^2. 

Here’s the kicker, Babylonian tablets from around 1800 BC have been found that show how to use the Pythagorean Theorem. It was used in surveying land boundaries more than a thousand years before Pythagoras lived. 

I would like to introduce you to Don’s Principle. “The apparent loss in mass of a submerged object is due to the mass of volume of the fluid displaced.” That is why things feel lighter when in water. Sure, some people call that Archimedes’ Principle. But it’s been over a thousand years, so let’s start calling it Don’s.

The Pythagorean Theorem has been around for thousands of years, used by surveyors, builders, and anyone who needs something to be square. It’s simple and reliable. You always trust that it works.

Honesty and responsibility are a lot like that, too. They’re not always easy in the moment, but they hold things together over time. Ignore them, and things start to drift. Use them, and everything lines up the way it should.

* “The sum of the square roots of any two sides of an Isosceles triangle is equal to the square root of the remaining side. Oh joy! Oh rapture! What a brain!” The Scarecrow in the Wizard of Oz stated this once he was awarded his ThD (Doctorate in Thinkology). I thought it was a mistake the writers made until doing a little internet detective work. It seems that it was an intentional mistake. The Scarecrow was already smart enough and getting a diploma made him no smarter.