how can I prove that pi times radius squared is the area of a circle

Look at a regular polygon of n sides.

The area is given by

$$A_n=\dfrac s 2 a n$$

s is the length of a side

a is the length of the apothem

n is the number of sides

I leave it to you to work out that for a given radius r

$$\dfrac s 2 = r \sin\left(\dfrac \pi n\right)$$

$$a=r \cos\left(\dfrac \pi n\right)$$

so

$$A_n=r^2 n\sin\left(\dfrac \pi n\right)\cos\left(\dfrac \pi n\right)$$

You can show that

$$\displaystyle{\lim_{n\to\infty}}n \sin\left(\frac \pi n\right)\cos\left(\frac \pi n\right)=\pi$$

so

$$\displaystyle{\lim_{n\to \infty}}A_n=\pi r^2$$

This limit corresponds to a regular polygon of increasingly many but smaller sides which taken to infinity represents a circle and thus you get

$$A_{circle}=\pi r^2$$

Thanks Rom,

That gives me several things to think about.  I had no idea how to do that one.

Plus, I have never heard the word 'apothem' before.

I have never seen that area formula before either but I am sure I could derive it easily enough.

How interesting!