How to determine the size of a SQL Server database for a specific date in the past?
Unless you capture that information on your own SQL Server does not record it anywhere. Most folks go by the backup size. 

If you use 3rd party software to backup the server or where ever you store your database backups you might see if it has a history as well. If you have something like Netbackup Enterprise or other similar software it should keep at most the file name and size unless your backup admins purge that information as well, but most don't that I have worked with.