Appendix to Lesson 23: The Volume and Surface Area of a Sphere
To obtain a formula for the surface area of a sphere, Archimedes imagined
the surface of the sphere divided into polygonal regions. Drawing radii as
suggested by the diagram in )LJXUH divides the volume of the sphere
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close to r, the radius of the sphere.
The volume of each cone is(^13) area of base .r
If B 1 , B 2 , ... , Bn are the areas of all the individual bases, then the approximate volume of the cone is
(^1133) Br 12 B r !!1 13 3B rnn B B 12 B r.
The true volume of the sphere is^43 Sr^3 , and this approximate formula approaches the true value if we use
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Now, B 1 + B 2 + ... + BnLVWKHVXUIDFHDUHDRIWKHVSKHUH6RZHFDQVD\WKDW
(^114333) BB 12 ! Brn surface area r|Sr (^3).
Algebra gives that surface area | 4 ʌU^2 ZLWKWKLVIRUPXODEHFRPLQJPRUHDQGPRUHH[DFWZLWK¿QHUDQG¿QHU
polygons used. To many, it seems reasonable to believe that the true surface area of a sphere can thus only
be 4ʌU^2.
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the leap to believe that the formula remains true in an “ultimate” sense
is the basis of calculus.
Historical comment: Archimedes was so proud of these results that the
diagram of a sphere inscribed in a cylinder was placed on his tombstone
at his request.
Figure 23.21
Figure 23.22