A famous problem that combines circles and lines is the "Broken Wheel" problem. It appears in many versions, where the wheel appears as a gear, or a ceramic plate, or maybe a vase. In any form, the problem is usually stated something like this.

Suppose you are out digging around in the ground and you discover a fragment of an old wheel (insert here any circular object of your choice). Of course you wish to know the size of the intact wheel , i.e. find its original radius.

Here is one way to calculate the radius. Put the fragment down on a piece of graph paper and approximate the locations of three points on the circumference. Call those three points P₁, P₂, and P₃.

Now the fun begins!

As an example, the red circular arc in this picture represents the wheel fragment. The three points that have been located are P₁ = (-2, -2), P₂ = (-1, 2) and P₃ = (2, 3) (the small red dots). The green line segments connect P₁ to P₂, and P₂ to P₃. Find the perpendicular bisectors (the blue lines) of the green line segments. Now just find out where the blue lines intersect. That's the center of the circle. (In this case it's at the point (35/22, -17/22).) Now the radius can be found by calculating the distance from the center to any of the three original points.

You can see the details of the calculations if you want, or you can try this example and make up your own problems using the Broken Wheel Fixer below. Just click on three points you want on your circle and then hit the bisect button.