Surface area is the measure of how much exposed area any two- or three-dimensional object has.
For a two dimentional object the area and surface area are the same:
square or rectangle: l * w (if square length equals width)
any regular polygon: P * a / 2 (where P = the length of the perimeter, and a is the length of the apothem of the polygon [the distance from the center of the polygon to the center of one side]).
a parallelogram: B * h (B [the Base] is any side, and h [height] is the distance between the lines that the sides of length B lie on)
a trapezoid: (B + b) * h / 2 (B and b are the lengths of the parallel sides, and h is the distance between the lines on which the parallel sides lie)
a triangle: B * h / 2 (where B is any side, and h is the distance from the line on which B lies to the other point of the triangle)
Some basic formulas for calculating surface area are:
cube: 6*(s2) , where s is the length of any side,
rectangular box: 2*((l * w) + (l * h) + (w * h)), where l, w, and h are the length, width, and height of the box,
sphere: 4*π*(r2) , where π is the ratio of circumference to diameter of a circle, 3.14159..., and r is the radius of the sphere.
An artist should feel free to add some example diagrams.