Fun With The Pythagorean Theorem

A few months ago I had a post about “You Should Get More of These” and I was talking about hinges. They let you build right triangles as shown in the picture below.

On my robot I am building now I need to make some cross braces and diagonal braces work best. I started tinkering around with the Pythagorean Theorem a bit to find a brace that would fit. I was hoping to find a brace length that would be close and if I had to “stretch” the braces a bit, I was fine with it. I created a large Excel spreadsheet using all the “A” and “B” lengths up to 50 and calculated the “C” length. I was surprised to learn that there are tons of right triangle combinations where the “C” length comes out to a whole number.

And of course you can use multiples of the numbers above such as a 6-8-10 triangle or a 10-24-26 triangle.  These numbers allow you to use even numbers when selecting a brace.  This is a piece of information that I will keep handy in the future.

OK, maybe it wasn’t THAT much fun.

Angled Beams

Have you ever wondered why these beams are at an angle of 143°? More precisely, 143.1301°.

It’s a good question, and I wondered for a long time as well. Then I discovered why, and it's actually pretty simple. It’s because of the Pythagorean Theorem. You know,

I discovered it while I was building a long truss, much like this one.

If you look closely at this simple truss and count not the holes but the hole spaces. The hole spaces are the length between the holes. For example, if you have a nine hole beam, there are eight spaces, a fifteen hole beam, there are fourteen holes spaces, etc. When you count them up, you'll find you have a common 3-4-5 right triangle.

Remember that the Pythagorean Theorem only works for right triangles, so the formula works out to be

3^2 + 4^2 = 5^2

or

9 + 16 = 25

I know that once I figured this out, life got much better and I began to use these oddball beams a whole lot more often.  The 3-4-5 right triangle is very useful in engineering and especially useful in building with Lego. I have used it several times, and I hope to share some constructions with you soon.