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 (where the base B is any side, and the height h 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). Alternatively, Heron's formula can be used: √(s×(s-a)×(s-b)×(s-c)) (where a, b, c are the sides of the triangle, and s = (a + b + c)/2 is half of its perimeter)
Some basic formulas for calculating surface areas of three dimensional objects 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
cylinder: 2×π×r×(h + r), where r is the radius of the circular base, and h is the height
cone: π×r×(r + √(r2 + h2)), where r is the radius of the circular base, and h is the height.
An artist should feel free to add some example diagrams.
If one adopts the axiom of choice, then it is possible to prove that there are some shapes whose area cannot be meaningfully defined; see Lebesgue measure for more details.
