Delete the entire back up of my iMac to make space on my air port time capsule

We are not sure which operating system that you might have installed on your iMac, but will assume that it is one of the more recent versions.

Use the Finder on one of your Macs to delete the backup file for the iMac.

Click on the Finder (smiley face icon), probably located on your Mac at the far left at the bottom of the Mac's screen

On the next window that appears, look for a listing of Locations on the left side panel of the window. On older systems this is labeled Devices

Click on the name of your Name Capsule under the Locations heading and another window will appear to the right

If you have not done this before, you will see a Connect As button at the upper right corner of the window

Click on Connect As and you will be asked for your Time Capsule administrator password

When you enter the password to connect, a folder named "Data" will appear in the window unless you have renamed the Time Capsule hard drive

Double-Click on "Data" to open the Time Capsule hard drive and display the contents

The next window will display files associated with the name of each Mac that has been backing up

Click on the name of the iMac file to highlight it

Click the Gear icon at the top of the window

Click Move to Trash

There is no going back at this point, so make sure that you have selected the correct name of the Mac backup file to be deleted. The deletion process can take a few minutes up to 10-15 or more depending on how much data is stored in the backup

Wait for the process to complete and you are all set.

One thing I have found with Bobs solution is that Finder and Time Machine can give different space availability, it's like Time Machine takes 3 days to catchup to the actual amount of space that is now available after deleting other backups. I