In the cooking of beef, the designation "medium" is generally agreed to have a warm, pink - not red - center and an internal temperature of 140° to 145°. Rare leaves the center cool and red (120° to 125°); medium-rare is warm and red (130° to 135°); medium-well is warm with little or no pink (150° to 155°); and well-done is uniformly grey-brown throughout (160° to 165°).
Over a hot grill, under the broiler, or in a medium-hot frying pan, a one-inch-thick hamburger will take about three minutes per side to reach rare, four minutes to reach medium, and five to thirty-seven minutes to reach well done.
While we're on the subject, the US Department of Agriculture wants you to cook hamburger to an internal temperature of at least 160°F (71°C). In other words, you are not to eat a hamburger that is not well done. Many people see no point in eating a hamburger that is well done. This is the definition of a conundrum.