I place the coin in a corner of the scanner (corner would vary depending on how you approach your scanner--for me it is the lower left corner based on reading the name brand EPSON) about an inch or 2 from the scanner margins and place a small piece of light colored paper or cardboard over the coin to provide contrast (and I think limits the size of the scan). Then I use the software that came with the scanner to crop the coin image, which usually enlarges it.
My scanners in the past usually made a pre-scan and I had the option to select a region to make a detailed scan which doesn't take as long as scanning the whole area
I use 1200dpi for a single coin but if I am scanning a set of coins, such as for a swap partner, then I reduce to resolution so as to manage the file size for sending. Also, on my scanner software, there is a facility to zoom and to adjust the scanning area down to the size of the object being scanned. If I am scanning a single coin, for example to add to the catalogue, then I use Photoscape (free to download) to crop and rotate the image.
Just because you can't see it ... doesn't mean it isn't there - Anon.
There is a lot missing in your question,
What type of computer?
What type of software?
I have a brother scanner with mac hardware and the default software is 'preview' with others available.
It's important that you choose to scan from the computer, and not by pushing a button on the scanner. The printer/scanner can be set to obtain an image and send to the computer, but it's best to start the image capture from the computer.
That way you can preview or fast-scan the entire field and then select the image to focus on..
Even if you are capturing the entire page field you should be able to crop it in your image processing program.
Again, what computer and program are you using? What does your picture look like after you scan it?