Cut the wooden block into a square measuring 7 inches by 7 inches by 4 inches, using a saw. Draw a circle on one side of the block using the compass. This will be the diameter of the dish. Cut away the wood outside of the circle with a jigsaw, or hacksaw if preferred, to leave a cylinder of wood.
Draw a second, smaller circle on the block, leaving three quarters of an inch between the inner and outer circles. This is the thickness of your wooden dog dish. Mount the block on a lathe. As the lathe spins, carve away the wood inside the circle with a cupped chisel until you have hollowed out the dish.
Sand down the inner and outer faces of the bowl using 120-grit sandpaper, to produce a smooth finish. Varnish the bowl using the paintbrush, to protect it. Allow the varnish to dry thoroughly, then varnish with a second coat. Allow this second coat to dry thoroughly before using the bowl for the first time.