Disclaimer: I didn't figure this out myself or even locate the procedure myself. However, I did decide that it would be nice to have it laid out step by step with some pictures.
Step 1: Gather the necessary supplies: Tape measure (preferably a soft one), bright colored duct tape, a notepad, a pen or pencil, a stopwatch, and a calculator. Or you can do what I did and use your iPhone/smartphone as a stopwatch, calculator, and camera.
Step 2: Measure your treadmill belt using the tape measure. I found that it worked best if I folded my treadmill up and then wrapped the tape measure around. Record your measurement in inches. (Mine was 112.75")
Step 3: Place a piece of brightly colored tape on the treadmill belt.
Orange worked well
Step 4: Set your treadmill to your normal running speed while you stand on the side rails or off to the side. (My treadmill takes a minute to get up to full speed so I waited to start timing until it was humming along nicely.)
10 minutes/mile according to the treadmill
Step 5: Wait for the tape to come around again and then start your timer. Start counting on the next appearance of the tape at the starting point (one rotation) and continue counting until you reach 50 rotations of the belt. Record the time it took.
Time to reach 50 rotations
Step 6: Do some math. The speed of your belt in MPH = (50 x length of your belt in inches x 0.0568) / the time you recorded.
(50 x 112.75 x 0.0568) = 320.21
320.21 / 51.3 = 6.24 MPH
So the 6 MPH button on my treadmill actually causes the belt to spin at almost 6 1/4 MPH! I conducted the experiment again at Shane's average running speed (8.0 MPH) and found that actually runs at 8.1 MPH!
So now I can either adjust my time/distance for what I know the actual pace is or I can remind myself that a 10 minute mile pace outside feels easier because it's ACTUALLY a 10 minute mile pace instead of 9:35 per mile. I'll probably stick with recording the treadmill's distance and time readout since I'm lazy but it's nice to know I'm not just being a baby about the hamster wheel!