A band saw is a power tool designed to cut wood with a long, ribbon-shaped blade that is most ideally suited for cutting straight lines. While a band saw is not the most intuitive choice for cutting circles, with a little ingenuity in how you design your cuts, you can create both positive and negative circles in wood.
Trace the circle you want in pencil on the wood, circumscribed inside a square. This means that the circle will be contained inside a square that's just large enough to hold it with the outside of the circle touching the outer edge of the square at four points (one on each side).
Cut out the square on the band saw.
Remove the corners around the circle. Use a straight edge to trace four cutting lines that will remove the corners but only touch the edge of the circle. This will leave an octagonal shape.
Continue cutting away the wood around the circle shape in straight lines. You will be creating polygons with more and more edges until the edges no longer appear to be straight lines and the circle shape emerges.
Sand the wood to refine the circle shape. Use a medium-grain sandpaper to remove the remaining polygonal corners, revealing a smooth, circular shape.
Trace the outline of a circle in pencil on a board. Use a compass, tie the pencil to a string, or trace a round object such as a tape roll in order to get a perfect shape.
Cut the board in half, bisecting the circle. Be sure to divide the circle into two equal halves.
Saw each half-circle into small slats. Trace the lines in pencil first, running parallel to the sides of the board on the narrow left and ride sides of the half-circle shape.
Break away the slats. If you position closely enough together, you should be able to break away the sections with your fingers. If not, cut a few more sections to thin the slats out.
Grind the inside of the half-circle shapes. Use a grinder or Dremel with a grinding attachment to remove the rough edges from the broken-off slats.
Reattach the two pieces of wood. If the board is small enough, simply use wood glue. If you need more force than that, use glue along with wood biscuits (small discs of wood that are inserted into slots cut into the edges of the boards) and a biscuit joiner for cutting the holes. Let the two pieces dry on a flat surface.