MySQL has the ability to show you size of the information contained in your database. You can find this information per database or per table in a database pretty easily.

Size of databases

To list the size of all the databases on the server run the following:

SELECT table_schema "Database", sum( data_length + index_length) / 1024 / 1024 
"Size in MB" FROM information_schema.TABLES GROUP BY table_schema;

Size of tables in database

To list the size of all the tables in an individual database run:

SELECT table_name, table_rows, data_length, index_length, 
round(((data_length + index_length) / 1024 / 1024),2) "Size in MB"
FROM information_schema.TABLES where table_schema = "{{NAME OF YOUR DATABASE}}";