Math CentralQuandaries & Queries


Question from Louise, a student:

i need to find the maximum volume of a cylinder that can fit inside a sphere of diameter 16cm

The volume of a cylinder is

V = pi r2 h             (1)

where r is its radius and h is its height. You need to differentiate this expression and then use what you know about calculus to find the values of r and h that maximize the volume. The challenge is that V is a function of two variables, r and h and you need to express V as a function of one variable to differentiate it. You need to use that fact that the cylinder is inside a sphere to eliminate one variable.


In the diagram C is the centre of the sphere and triangle ABC is a right triangle. Use Pythagoras' theorem to find a relationship between r2 and h2. Use this relationship to eliminate one variable in equation (1) and then use calculus to maximize the volume of the cylinder.

I hope this helps,

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