Find the IP address of a computer with know MAC address (ethernet ID)

So, on my server it is reporting an issue with a computer with MAC address AA:BB:CC:DD:EE:FF (obviously I made this one up).

Trouble is, I know this computers MAC address but not it's IP address. So how do you find it?

Well, found this forum and this forum which basically tells you to use the command 'arp':

arp -a | grep "AA:BB:CC:DD:EE:FF "

arp lists all MAC address of computers on your subnet. But it's taking a long time. Reading more in the forum it looks like it works by pinging all the IP addresses, then asking it for it's MAC address. Long winded!