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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 πr². As the cylinder has a height equal to the sphere, its volume must be πr² × 2r = 2πr³, and so...

Theorem
The volume of the sphere, V, is given by V = 4πr³/3, 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_k² = r² – H_k² = r² – (k/n)²r² = r²(1 – k²/n²)

Therefore R_k² = (r²/n²) (n² – k²)

The area of the base of the k th layer will be πR_k² = (πr²/n²)(n² – k²).

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 (πr²/n²)(n² – k²) × r/n = (πr³/n³)(n² – k²).

Therefore, if the volume of the sphere is V, the volume of the hemisphere, ½V, can be approximated.

`V`	≈ (π`r`³/`n`³)(`n`² – 1²) + (π`r`³/`n`³)(`n`² – 2²) + ... + (π`r`³/`n`³)(`n`² – `n`²)
	= (π`r`³/`n`³)((`n`² – 1²) + (`n`² – 2²) + ... + (`n`² – `n`²)
	= (π`r`³/`n`³)(`n` × `n`² – (1² + 2² + ... + `n`² ))
	= (π`r`³/`n`³)(`n`³ – (1² + 2² + ... + `n`² ))

Using the result
1² + 2² + 3² + ... + n² = ¹/₆ n(n + 1)(2n + 1)

`V`	≈ (π`r`³/`n`³)(`n`³ – ¹/₆ `n`(`n` + 1)(2`n` + 1))
	= (π`r`³/6)(6 – (1 + 1/`n`)(2 + 1/`n`))

As n tends towards infinity, 1/n tends towards zero and so ½V tends towards (πr³/6)(6 – 2) = 2πr³/3.

Hence the volume of a sphere, V = 4πr³/3.