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Square Root Drawing Lines and a Circle in AutoCAD

by Carol Dunn

*Note: If you just want to see the video, scroll to the bottom of the page*

I’ve always been fascinated with what you can do with a simple straight edge and a compass. The math of the Ancient Greeks used these two tools, and that’s it. Think of it as using a standard drafting compass and a triangle with no markings on it. What kind of math could you do with just those two things? Well, there’s actually quite a bit you can do!

A few examples of things you can do, you can add two numbers, you can subtract two numbers, you can multiply, you can divide, you can square a number. You can even find the square root of a number!

When I heard that you could find the square root, I thought to myself, there’s no way! How would you do that? Well, it turns out to not be too terribly difficult, once you know how.

I’m going to be using AutoCAD in this demonstration but you can use any program you like, or heck, even a piece of paper, a compass, and a straightedge!

Step 1, draw a line with a length representing the number you want to find the square root of. In this example, I’m going to use the number 16 (because it’s easy).

Next, draw another line starting at the end of this line, extending in the direction of this line (i.e. co-linear), a unit length.

Now, draw a circle using the left hand endpoint of the first line and the right hand endpoint of the second line as it’s diameter.

Finally, draw a line vertically from the right hand end of the original line to the circle. The length of this line is the square root of the original line!

HERE is a video of me doing this same procedure using a slightly different number.

Now, for those of you that are the curious math type, why is this the case? Can you figure out why this does this?

Stay tuned for examples on why this works the way it does!