Circumscribe a Polygon

A polygon circumscribed about a circle has the midpoints of all of its sides lying on the circle. 

Example 1:  The following is an example on how to construct a square circumscribed about a given circle provided you can rotate ninety degrees.

 Given the circle with Center O construct the square that is circumscribed about the circle.

Construct the line segment OP that is the radius of circle O.

Rotate line segment OP 90 degrees about the center O.  Repeat this step three times such that there are four points that lie on the circle.

Construct the lines that are tangent to the circle at these four points. These tangents are perpendicular to the radii.  These lines will intersect at four points.

Construct the line segments that connect the four points. This is the square that is circumscribed about the given circle O.

Example 2:   The following is an example on how to construct a regular pentagon circumscribed about a given circle provided you can rotate seventy two degrees.

Given the circle with Center O we will find five points that are equally distributed on the circumference of circle O.

Construct the line segment OP that is the radius of the circle O.

Rotate line segment OP 72 degrees about the center O.  Repeat this step four times such that there are five points that lie on the circle.

Construct the lines that are tangent to the circle at these five points.  These tangents are perpendicular to the radii. These lines will intersect at five points.

Construct the line segments that connect these five points. This is the regular pentagon that is circumscribed about the given circle O.

Example 3: Euclidian Construction of an circumscribed Pentadecagon

I  have previously showed how to construct a 24 degree angle on the Construction of a 24 degree angle page. 
To construct a pentadecagon you must construct fifteen 24 degree angles.  We then construct lines perpendicular to the points where the angles intersect the circle. The first intersection of these lines is one vertex of the pentadecagon.   

We then connect the fifteen vertices with line segments. This is an example of a regular pentadecagon circumscribed about a given circle.

Example 4:  The following is an example on how to construct a regular decagon circumscribed about a given circle.  Since ten equals five times two, we can construct a pentagon and then bisect the central angles of the inscribed pentagon.  The points where the angle bisectors intersect the circle are the remaining five vertices of the decagon.  

Construct a five n-gon. 
Bisect the angles such that each angle will now measure 36 degrees.  
Construct the lines that are tangent to the circle at these ten points.  Construct points at the first intersections of these lines.  
Construct the line segments that connect these points.  These line segments will form the sides of a regular decagon that is circumscribed about the given circle O.