Geometry: 3-D Measurements
Surface area measures the area of a surface--essentially it is the same as area. The unit of measure for surface area is the square unit, just as in area. However, the measure of surface area becomes troublesome when we try to calculate the surface area of figures whose surface or surfaces aren't regions in a plane. In these cases, multivariable calculus is sometimes necess ary. In this text, we'll focus on calculating the surface area of polyhedra and spheres, surfaces we know we can understand and utilize without resorting to calculus.
Surface Area of a Polyhedron
The surface area of a polyhedron is the sum of the areas of the polygons that compose the polyhedron. The only special formulas for surface area of polyhedra are extensions of those for particular polygons: certain shortcuts become possible when the comp onents of a polyhedron are special two-dimensional figures that we've already studied. For example, the surface area of a right prism whose bases are regular polygons is four times the area of any lateral face, and two times the area of either base. Thi s is true because the lateral faces are congruent to one another, and so are the bases. The simplest way to calculate the surface area of a polyhedron, though, remains to simply sum the areas of the polygons that make up its faces.
Surface Area of a Sphere
The surface area of a sphere has a very interesting formula. It depends solely on the radius of the sphere. The surface area of a sphere is equal to 4Π times the square of the radius of the sphere: 4Πr 2 . This formula can be derived by thinkin g of the sphere as a polyhedron consisting entirely of pyramids sharing the center of the sphere as their vertex. As the area of the base of such pyramids decreases, the surface more closely resembles a sphere. This just goes to show that by using formu las we already know, we can derive the formulas for various surface areas.