If the diameter of a circle is double the diameter of a second circle, what is the ratio of area of the first circle to the area of the second?

Let's look at the formula for the area of any circle:

               Area = (pi) times (RADIUS SQUARED) .

I made that last part LOUD AND HEAVY because that's where
the answer to this question is.

Notice that the area depends on the square of the radius.
I won't go into a lot of detail to prove this, but that means that
if you increase the circle's radius by some factor, then you
increase the circle's area by the square of that factor.

So if the other circle has double (2 times) the diameter,
then it has  (2-squared) = 4 times the area.