This instructable shows how to find the center of a circle. Many, many projects involve circular things and a need to easily find the middle. For various reasons pre-determining the center of a cut circle can be difficult, particularly when it will affect balance in motion.

I was working on the Cardboard Wind Turbine project and the most convenient large diameter template wasn't conveniently sized for pre-measuring and cutting out a square and then matching the template to it seemed tedious and error prone. Also I've never been very good at drawing on the lines. So here we are.

I'd like to thank and acknowledge the folks at Math Open Reference for their contribution.

So what we're going to do is work with a cardboard circle I've cut out, find the center point and then create a "crosshairs" so that work on various projects can proceed smoothly.

Step 1: Bill of Materials

Here's a list of items we're going to be using.

Carpenters Square or other right angle equal to or larger than the diameter (size) of the circle. I've got a fancy, schmansy carpenters square that I finally found a use for but any reasonably stable right angle will. I've used both the carpenters square and a large manila envelope to demonstrate the utility of the technique.

A compass - I show a basic protractor and compass set available at any drugstore, supermarket or department store. The protractor (the half circle thing) isn't actually used. It is possible to substitute the traditional pencil and string compass but I do not demonstrate that technique. Save yourself a great deal of trouble and spend the two dollars, over time you will thank me.

The box cutter and cardboard box isn't required unless you don't have a circle handy to practice on.

I show two cardboard circles of approximately equal size that I need to find the centers of so I place them on an axle and have them rotate at fairly high speeds so the better the balance the more stable my contraption. Any circle may be substituted.

Step 2: Finding the Center of the Circle

So how do we find the center? Our solution to this problem lies in a basic law of geometry called Thale's Theorum what this says is that any line of drawn across the diamater a circle always includes a right angle to some point on the circle

So if we place a right angle on the edge of the circle and we mark the spots at which the angle intersects the edge we will have identified a line across the diameter of the circle.

Place the right angle on any point on the edge of the circle so that the legs extend over the edge.

Mark the exit points on both legs of the right angle with a line the edge until it exits the circle.

Use the ruler to draw a line that connects the two points at which the lines exit the circle. This is a diamater line.

Now repeat this process for any other point on the edge of the circle (do not attempt to find the 'right' angle to make the crosshairs, in the next step we will divide the circle by the other half).

The point at which the two lines intersect mark the center of the circle. If that's all you need to find then we're done. Otherwise in the next step we will use our compass to create the classical 'cross hairs' that often needed for further work involving circles or cylinders.

If we do this for a second line the point at which the two diameter lines intersect is the middle.

Step 3: Creating the Crosshairs

Now that we've found the center in one direction we're going to identify the center in the other direction.

First we have to pick one of our diameter lines to serve as the X axis. We will construct a perpendicular line from the center to create the Y axis and extend it to both sides.

So we're going to take our compass and set it to approximately one half the radius of the circle (or about halfway between the center and the edge).

Placing the point of the compass on the center we going to mark the point at which the compass interects the X axis.

Next we're going to place the point of the compass on each of these and mark where the two circles overlap. It is not necessary to draw the entire circle (or even a very big arc). Just big enough to detect where the two circles would intersect.

Now using the ruler or other straightedge we will draw a line between the intersection and the middle. This is the Y axis line.

Mark the X and Y axis so you don't forget and you are ready to proceed with your project.