Frequently Asked Questions
What is the volume of a sphere?
Archimedes discovered, and proved, that a sphere occupies precisely two-thirds of the cylinder that contains it.
If the cylinder has a radius, r, the area of the base is πr^{2}. As the cylinder has a height equal to the sphere, its volume must be πr^{2} × 2r = 2πr^{3}, and so...
Theorem
The volume of the sphere, V, is given by V = 4πr^{3}/3, where r is the radius.
Proof
Let us consider the top half of a sphere, radius r.
We shall say that the radius of the k th layer is R_{k} and if we split the height of the hemisphere into n layers, the k th layer will have a vertical height of H_{k} = (k/n)r.
Using the Pythagorean Theorem,
R_{k}^{2} = r^{2} – H_{k}^{2} = r^{2} – (k/n)^{2}r^{2} = r^{2}(1 – k^{2}/n^{2})
Therefore R_{k}^{2} = (r^{2}/n^{2}) (n^{2} – k^{2})
The area of the base of the k th layer will be πR_{k}^{2} = (πr^{2}/n^{2})(n^{2} – k^{2}).
The height of each slab will be r/n, so the volume of the k th slab (assuming it to be a cylinder) will be (πr^{2}/n^{2})(n^{2} – k^{2}) × r/n = (πr^{3}/n^{3})(n^{2} – k^{2}).
Therefore, if the volume of the sphere is V, the volume of the hemisphere, ½V, can be approximated.
V | ≈ (πr^{3}/n^{3})(n^{2} – 1^{2}) + (πr^{3}/n^{3})(n^{2} – 2^{2}) + ... + (πr^{3}/n^{3})(n^{2} – n^{2}) |
= (πr^{3}/n^{3})((n^{2} – 1^{2}) + (n^{2} – 2^{2}) + ... + (n^{2} – n^{2}) | |
= (πr^{3}/n^{3})(n × n^{2} – (1^{2} + 2^{2} + ... + n^{2} )) | |
= (πr^{3}/n^{3})(n^{3} – (1^{2} + 2^{2} + ... + n^{2} )) |
Using the result
1^{2} + 2^{2} + 3^{2} + ... + n^{2} = ^{1}/_{6} n(n + 1)(2n + 1)
V | ≈ (πr^{3}/n^{3})(n^{3} – ^{1}/_{6} n(n + 1)(2n + 1)) |
= (πr^{3}/6)(6 – (1 + 1/n)(2 + 1/n)) |
As n tends towards infinity, 1/n tends towards zero and so ½V tends towards (πr^{3}/6)(6 – 2) = 2πr^{3}/3.
Hence the volume of a sphere, V = 4πr^{3}/3.