If we don’t care that the circle has to be perfectly round, and “Almost” round will do, then the math is really simple.

Any grade school kid can make a circle with a piece of string, a pin, and a pencil. The circle’s radius is the length of the string. I’m sure you remember the process, here’s a picture.

Now, instead of the string, let’s take an imaginary string that’s just the radius of the circle, anchor one end at the center of the circle, and rotate it a degree at a time.
The X will be the horizontal line, and the Y will be the vertical line. Starting at the bottom, the first point of our plot will be x 5″, y 0″.

As always, click to embiggenate.

Now the nine red lines represent nine individual movements of our “String” of one degree each. The copy paste from the spreadsheet, below, will show what each line means. (Clarification: The spreadsheet solves for the endpoint of the red line, so we can get those line segments that make the “Circle” made up of 360 line segments.)


You can see that the first line, one degree away, has only a very tiny motion from zero in y. And the amout it moves in from X is also very tiny. But the more degrees you move, the closer the Y gets to 5, and the closer the X gets to zero. At 45 degreees, shown in green on the plot, the X and Y distance are exactly the same; at 90, the x becomes zero and the Y becomes 5″.

Anyone still with me?