Show, how given a distance r, one can construct a regular pentagon, whose circumradius is r, using only ruler and compass...

Note 1: The ruler doesn't have any marks, so it's no good for measuring. It's only good for joining 2 points by a line.

Note 2: The circle which circumscribes the polygon, such that the polygon lies entirely within the circle and all of whose vertices lie on the circumference of the circle is the circumcircle of the polygon. The radius of this circle is the circumradius of the circle.

Start with a line. Erect a perpendicular to it. Set your compass to length r. Measure off this distance on one of the perpendicular lines, and twice this distance on the other. Connect the endpoints. You've formed a right triangle with hypotenuse √5 in these units of r.

Using your unchanged compass mark off 1 unit (r length) from the hypotenuse, producing a line of length (√5)-1 in these units. Then bisect this remaining length, and bisect one of the two halves again, using the standard technique of erecting a perpendicular bisector. You now have a line segment of length ((√5)-1)/4 times r and a perpendicular to it at one end. At the other end place the compass point and draw a circle, which intersects the perpendicular line. Connect the center of the arc to this intersection point. At the center of the arc you will have created a 72-degree angle.

You have drawn a full circle of radius r around the 72-degree angle vertex as a center.

Now fit the compass to the two points on the circle intersected by the two lines from the 72-degree angle and mark of four more of these 72-degree arcs (three more points actually). Connect the endpoints of the 72-degree arcs. The result is a regular pentagon with a circumradius of r.

Posted by Charlie on 2003-05-17 09:08:11