Divide a circular disk into seven parts with a straightedge and compass such that each part has the same area and perimeter.

For some reason this seems intuitive to me.

Start by determining the center of the disk using the compass. I suppose at a rudimentary level, this can be done by trial and error with the compass, drawing test circles until you've exactly traced the disk. Once you have the center, draw the diameter of the circle.

Now, you need to divide the diameter into seven equal parts. This can be done with a straightedge and compass. You need to extend a line from one endpoint of the diameter to some point in space. The line should be a length of 7x, where x is some spacing of the compass. Divide this line into 7 segments with the compass (assuming you hold it steady). Construct a similar line rotated 180 degrees from the other endpoint, and create 7 equally spaced segments on this line. These two lines should be identical, in dimenions, just rotated 180 degrees and extending from opposite sides of the diameter. Connecting the corresponding points on these rays creates a segment that crosses the diameter - connect 6 points and you'll have 7 equal segments of the diameter.

Now, using the compass, bisect each of these segments. You now have 14 equally spaced points on the circle's diameter. Number these from 1-14, with 0 representing the first endpoint and 14 representing the other endpoint. Place the compass at point 1, and draw a semicircle from point 0 to point 2. In all, draw semicircles with the following centers and endpoints: (1,(0,2)), (2,(0,4)), (3,(0,6)), (4,(0,8)), (5,(0,10)), and (6,(0,12)). Now you get 6 semicircles of increasing size on one half of the disk. Rotate the disk 180 degrees and perform the same operations to complete the 7-section. The largest portion on the top corresponds to the smallest portion on the bottom, and so on.

Posted by Eric on 2004-12-09 17:35:53