Making the circles or the holes is a good trick to have in your mental tool box.
#1 Make a wooden (I use 3/4" ply as I can attach extensions onto that ply) rectangle that is about 20" long and wide enough to mount your router to. (from this time on we will call this wooden piece the Plate).
#2 Put a mortising bit in your router, 3" X 1/2" shank with a 1/2" d. cutter is what I use in this plunge router.
#3 Attach your router to the Plate at one end.
Wear your Safety Glasses
#4 Make sure the router is secure and the bit is in tight, now plunge the 1/2'' bit down through the Plate that the router is attached to.
#5 With the router turned off, adjust the bit depth almost flush with the bottom of the Plate, so it is just sticking out through the bottom of the Plate about 1/8".
If you want to cut a 15" hole;
#6 Lay the router and Plate on their side with the router on the right and the majority of the Plate to the left.
#7 Using a measuring device, start at the right side of the cutting bit, measure to the left 7 1/2" and make a mark. This will be the distance of half of a 15" hole. I use a Lufkin X46 6' Red End in this photo.
#7a. Take a #6 (2") bright finish nail and cut off the head with your side cutters or what ever you use.
#8 Insert the headless nail into the chuck and drill/burn a hole through the 3/4" plate at the place marked (7 1/2"). Try your best to go straight through with your drilling. Don't force the nail just apply reasonable pressure.
#9 When drilling is complete retract the mortising bit so that the plate will sit flush on the table.
#10 Take a new headed #6 bright finish nail and gently tap it into the pre-drilled/burned hole in the Plate, with the point headed toward the bottom of the plate.
#11 Get the piece you are intending to cut the hole into and consider how you can cut all the way through the piece without ruining something good supporting it. I usually elevate it on 3/4" sacrificial strips that stick out on the ends so I can secure the strips to my table with clamps or screws. If you use screws put none under the piece you are routing unless you are very sure of their placement and the routing path.
#12 Secure the piece with screws out side the finished hole and if you don't intend to use the circle secure it with screws too.
#13 Determine the center and make a mark.
#14 With and awl or a nail make an indentation for a starting point.
#15 Raise the Plate and tap the nail through the Plate at the 7 1/2" mark so about 1" of the point is sticking out. (this makes it easier to get the nail centered on the piece you want to cut out.)
#16 Set the nail into the indentation and tap it in securely but not all the way through the piece you will be cutting.
#17 Swivel the router around in a counter clockwise manner to see what obstacles may hinder your path.
#18 Turn on the router, allow it to get up to speed, then plunge down about 1/4'' secure to that depth and revolve the plate and router to cut the first pass.
#19 When completed with the first pass, plunge a bit deeper and secure at depth, revolve again.
#20 Take your time and repeat plunging until all the way through.
#21 Your results should leave you with a 15" hole and a 14" circle.
What ever the results are, mark the nail hole on your Plate with the exact finished dimensions it cut (measured after you are finished) so you have that number accurately recorded.
Next ...The circle to fit the hole.