How do I make a circular frame?

I'm making a large circular frame to trim out a round window in an old house. The outside diameter needs to be 22", and the inside diameter 14". I would like to use the same 5"-wide board so the grain is continuous. Given the dimensions of the circle and the lumber, how do I figure out how many segments to make, and at what angle is each to be cut?
—Craig Vanaken, Aloha, Ore.

The more segments you combine to make a circle, the smoother the grain appears to flow between segments, Craig. For the diameter of your circle, eight segments should be enough. Also, this makes the necessary 22.5° miter angles easy to cut.

First, on a 2'-square piece of paper, use a compass to draw two concentric circles with 11" and 7" radii. Then draw a line through the center and use a framing square to draw a perpendicular line dividing the circle into quarters. Using either the compass or a ruler, divide each quarter in two for a total of eight equal sections.

Because you don't want to risk ruining your lumber, practice on a 5"-wide strip of ¾" plywood to fine-tune your miter gauge setting and assembly technique. Use the circle to determine the final outside length of each section, and cut your board slightly oversize. (In this case, we cut 10" blanks for 9½" sections.) With a miter gauge and auxiliary extension, cut 22.5° angles on one end of each piece. Then, set a miter gauge stopblock to cut the sections to final length.

Assemble two half-circles of four pieces each, like the one shown above, using biscuits for alignment, and tape to hold the pieces tightly together.

Next, cut your paper pattern in half, and clip out the half-circles to within about ½" of the inside and outside lines. Spray-adhere a half-pattern to each half-circle glue-up. Bandsaw and sand both to the pattern lines, leaving just a little extra on the inside and outside of the ends. If your miter gauge is misaligned, you may need to file or sand the ends of the half-circles for a tight fit. Glue the two half-circles together, then sand the joints smooth.