Constructing a wooden octagon planter requires cutting precise 22 1/2-degree angles at the ends of the boards that will form the planter's eight sides. That task would be nearly impossible when estimating the angles while cutting the boards, but a miter saw can be set to the exact angle to make a perfect cut each time. A circular saw used with a miter box works similarly. Drainage holes are needed in the planter's bottom to let water out. Choose naturally rot-resistant lumber such as redwood to avoid the use of wood containing pressure-treating chemicals.
Cut lumber to make eight board pieces for the wood planter's sides, making the length of each side roughly one-half the desired diameter of the planter. Make straight cuts at each end of the wood pieces. The size of the lumber to use depends on the height you desire for the planter. Use 2-by-6-inch lumber, for example, to make a planter with an actual height of 5 1/2 inches. A planter made from 2-by-12-inch lumber will have an actual thickness of 1 1/2 inches and an actual height of about 11 1/4 inches.
Adjust a miter saw or miter box to 22 1/2 degrees. Cut each end of the eight board pieces at a 22 1/2-degree angle, making the inside edge of each board slightly shorter than the outside edge. You might be able to stack and cut several boards at one time, depending on your saw.
Arrange the eight angle-cut boards on their narrow edges to form an octagon. Notice where the boards' ends meet each other. Disassemble the octagon. Apply a 1/4-inch-wide bead of wood glue along the board ends that met each other in the octagon. Reform the octagon, and push the boards together tightly at the octagon's corners to hold them together.
Drive 1 1/2-inch-long wood screws inward at each corner, securing the boards together. Use a minimum of two screws at each joint -- one at the top and one at the bottom, or use three screws for wide boards, such as 2-by-12-inch boards. Drive each screw in at an angle so it goes into both boards at the corner and so the screw's end doesn't protrude through the back of a board.
Set the assembled octagon on top of a piece of 1/2- to 3/4-inch-thick plywood sheet or medium-density fiberboard, lining up the octagon with an edge of the plywood or fiberboard. Draw around the octagon with a pencil, transferring the octagon's size and shape to the plywood or fiberboard. Cut out the octagon shape on the plywood or fiberboard by using a circular saw.
Apply a thin bead of wood glue on the octagon's narrow edge that you want to be against the planter's bottom. Align the plywood or fiberboard octagon shape on top of the narrow edge you glued, and press the octagon shape firmly against the narrow edge for 1 to 2 minutes, helping the glue bond. Stop pressing the octagon shape, and wait about 30 minutes for the glue to dry. The plywood or fiberboard octagon will be the planter's bottom but should face upward for this part of the construction process.
Drive 1-inch-long wood screws through the planter's bottom and into the narrow edge of the assembled wood octagon. Use one screw just outside each of the eight board joints.
Drill a 1/2-inch-diameter hole through the center of the planter's bottom, using a power drill and a 1/2-inch-diameter drill bit.
Paint or stain the wood octagon planter your choice of color. Alternatively, leave the planter its bare wood color but paint it with a clear water sealer to help protect the wood.