Surface Area of A Sphere
The surface area of a sphere is given by the following formula:
This formula was first derived by Archimedes, based upon the fact that the projection to the lateral surface of a circumscribed cylinder (i.e. the Lambert cylindrical equal-area projection) is area-preserving. It is also the derivative of the formula for the volume with respect to r because the total volume of a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness.
At any given radius r, the incremental volume (δV) is given by the product of the surface area at radius r (A(r)) and the thickness of a shell (δr):
The total volume is the summation of all shell volumes:
In the limit as δr approaches zero this becomes:
Since we have already proved what the volume is, we can substitute V:
Differentiating both sides of this equation with respect to r yields A as a function of r:
Which is generally abbreviated as:
Alternatively, the area element on the sphere is given in spherical coordinates by . With Cartesian coordinates, the area element . More generally, see area element.
The total area can thus be obtained by integration:
Read more about this topic: Sphere
Famous quotes containing the words surface, area and/or sphere:
“A novelist is, like all mortals, more fully at home on the surface of the present than in the ooze of the past.”
—Vladimir Nabokov (18991977)
“... nothing is more human than substituting the quantity of words and actions for their character. But using imprecise words is very similar to using lots of words, for the more imprecise a word is, the greater the area it covers.”
—Robert Musil (18801942)
“One concept corrupts and confuses the others. I am not speaking of the Evil whose limited sphere is ethics; I am speaking of the infinite.”
—Jorge Luis Borges (18991986)