Octagonating a Square

Welcome to the math questions blog!

On this blog I will publish interesting questions asked on Yahoo! Answers and I will try my best to answer them as clearly as possible. Also, if you post questions on the comments section, I will try my best to answer those too.

So, to start with, we have a questions about making an octagon from a square:

A square table is 48 inches on each side. You want to re-shape that table into an octagon, and you're going to do it by cutting off the corners. How long will each side of the octagon be?

At first you might be tempted to divide each side of the table by 3 and call it the answer, but you are cutting a diagonal piece of an isosceles right triangle, which means we're dealing with square roots here. So how do we solve that?

First, we call the length of the cut (the diagonal cut), which is also the length of the new octagon side, x. We are cutting an isosceles right triangle off each corner, and their hypotenuse is x (as we defined it). That means the length of the triangle's leg is x / sqrt(2). New term - sqrt means square root (of what's in the parentheses).

Now we know that each side of the table is twice x / sqrt(2) + x. That's because each side of the table has two "triangle legs" and the length of the side of the octagon. The side of the table is also 48 inches long, and that's where we make our equation:
2x / sqrt(2) + x = 48

When we solve that equation, we get that x = 19.882...

So there you have it: if you have a square table with a side length of 48 inches and you wanted to make a regular octagon out of it, every side of the octagon should be approximately 19.882 inches long.

Also, if you want some nice math tricks and tips, visit the Super Math Tips and Tricks page.

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Yours,
Nadav

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