This plugins might help you. Not exactly by giving you the area by enetring coordinates, but it makes measuring areas faster.
1-CalcArea: this plugin calculates the area of a polygon while it's being created or edited
2-AutoFields: Automatic attribute updates when creating or modifying vector features
This plugin allows you to configure vector fields to be automatically calculated for you when digitizing new features or modifying existing ones. Think about it as triggers in a database or as an automatic Field Calculator.
I personally use the second one. When I import a shapefile, i activate the plugin to automatically generate an Area attribute (new or existing column), and automatically calculates the Area of all existing polygons. If i add or edit poligons in that shapefile, the area is re-calculated automatically.
When there is an "Area" attribute in your shapefile, then the "identify tool" can be used to see the polygon´s area with one click.