Make the needed measurements for the hole. Determine where the hole will go on the counter. Measure the diameter of the item (such as a faucet) to be placed over the hole. The hole only needs to be slightly larger than the pipe or tube running through it; make sure the hole is smaller than the diameter of the item that will cover the hole.
Place a chalk mark just above and below the hole location. Place a long strip of masking tape over the area to be drilled. Draw a chalk line on the tape connecting the two other chalk marks. Place a small "X" in the middle of the third chalk line.
Drill a small starter hole on the "X" using a spiral drill bit. The bit should be a little smaller than the router or hole-saw bit. Drill the hole all the way through.
Attach the router or hole-saw attachment to the drill bit. Position it over the starter hole; the starter hole should be in the center of the bit.
Press the drill button in about halfway to get the drill bit spinning at a medium speed. Do not run the drill at full speed.
Press down gently on the drill while you drill through the Corian surface. Move slowly through the hole in a smooth motion, and do not release the drill button while drilling.
Remove the masking tape. Sand down the edges around the hole and inside the hole with sandpaper until smooth. Repeat these steps for each additional hole you need to drill.