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
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