As Uncle and Ronny mentioned, Corel doesn't do that unless you an appropriate macro.
For permitting- unless it's a shape where you actually can get the area via a geometric equation (a circle, rectangle, or triangle, for instance)- we do the "draw a rectangle around it" (or, for particularly tight restrictions, draw the rectangle around the elements) and figure the area accordingly.
Not what you asked about, but I'd mention one thing Corel does do that I think is pretty handy is give you the perimeter of items- which I use for determining how much material I need for jobs (particularly material for framing, channel letter coil, etc.). To get that number, convert your shape/letter/whatever to curves and go to Properties/Curve and it will tell you the length of the curve (the perimeter) of your item.