Constructing a Regular Pentagon

In this assignment we will construct a regular pentagon with the methods of ruler and compass constructions. The pentagon will be constructed by first drawing a circle S which will circumscribe the pentagon. We will also construct a regular octagon. The instructions for the pentagon are fairly intricate, so read carefully the directions.

Constructing a Regular Pentagon:

1. Draw a line segment OB (make it about 1 inch long) in the middle of a piece of paper. The point O will be one vertex of the pentagon.
2. Draw the perpendicular bisector to OB, and label the midpoint of OB as A. You may wish to recall the procedure for doing this from the assignment Ruler and Compass Constructions. Next, draw enough of the circle centered at A whose radius is the length of OB to see where it crosses the perpendicular bisector. Label the two points of intersection as C and C'. You will need to draw your bisector long enough so that the circle does cross it twice.
3. Draw the circle centered at C passing through O. This will be the circle that circumscribes the pentagon you will draw. Label the circle S.
4. Draw the line segment OC, and draw enough of the circle centered at O and passing through A to see where it intersects OC; call the intersection point D.
5. Draw the line L perpendicular to CD and passing through D. You can repeat the method of drawing bisectors again, or use the 90 degree angle of your triangle to do this.
6. Draw enough of the circle centered at C and passing through C' to find its intersections with L. Label the intersection points E and F. Make sure you drew L long enough to have the circle cross L in two points.
7. Draw the line segment CE and mark the intersection E' of the circle S with CE. Similarly, draw the line segment CF and mark the intersection F' of S and CF. The points E' and F' will be vertices of the pentagon.
8. Draw enough of the circle centered at E' passing through O to find the point of intersection G of this circle and the circle S. This point will be the fourth vertex of the pentagon. Also draw enough of the circle entered at F' passing through O to find the point of intersection H of this circle and S. This point will be the final vertex.
9. Connect the five vertices O, E', G, H, and F' to get the pentagon.

If you wish to see all nine steps at once, click here.

Constructing a Regular Octagon:

1. Draw a line segment AB
2. Draw a perpendicular bisector to AB, and label the midpoint C. You can do this step by using step 2 of the procedure for constructing a pentagon.
3. Draw the circle S centered at C and passing through A (that also passes through B). Label as D and E the points where this circle crosses the perpendicular bisector you drew in the previous step.
4. Draw enough of the circles centered at B and at D (with the radius of the circle in the previous step) to find the point outside of the circle S where these two circles cross. Draw the line passing through this intersection point and C, and draw it long enough to mark the two points where it crosses the circle S.
5. Repeat this step with points A and D. Doing so will mark two more points on the circle S.
6. You now have eight points on the circle. Connect them to get the octagon.