Surface Area of A Sphere
To find the surface area of the sphere, Archimedes argued that just as the area of the circle could be thought of as infinitely many infinitesimal right triangles going around the circumference (see Measurement of the Circle), the volume of the sphere could be thought of as divided into many cones with height equal to the radius and base on the surface. The cones all have the same height, so their volume is 1/3 the base area times the height.
Archimedes states that the total volume of the sphere is equal to the volume of a cone whose base has the same surface area as the sphere and whose height is the radius. There are no details given for the argument, but the obvious reason is that the cone can be divided into infinitesimal cones by splitting the base area up, and the each cone makes a contribution according to its base area, just the same as in the sphere.
Let the surface of the sphere be S. The volume of the cone with base area S and height r is, which must equal the volume of the sphere: . Therefore the surface area of the sphere must be, or "four times its largest circle". Archimedes proves this rigorously in On the Sphere and Cylinder.
Read more about this topic: The Method Of Mechanical Theorems
Famous quotes containing the words surface, area and/or sphere:
“And yet we constantly reclaim some part of that primal spontaneity through the youngest among us, not only through their sorrow and anger but simply through everyday discoveries, life unwrapped. To see a child touch the piano keys for the first time, to watch a small body slice through the surface of the water in a clean dive, is to experience the shock, not of the new, but of the familiar revisited as though it were strange and wonderful.”
—Anna Quindlen (b. 1952)
“The area [of toilet training] is one where a child really does possess the power to defy. Strong pressure leads to a powerful struggle. The issue then is not toilet training but who holds the reins—mother or child? And the child has most of the ammunition!”
—Dorothy Corkville Briggs (20th century)
“For my part, I have no hesitation in saying that although the American woman never leaves her domestic sphere and is in some respects very dependent within it, nowhere does she enjoy a higher station . . . if anyone asks me what I think the chief cause of the extraordinary prosperity and growing power of this nation, I should answer that it is due to the superiority of their woman.”
—Alexis de Tocqueville (1805–1859)