Calculate the area of a circle with a radius of 2 cm and a circle with a radius of 4 cm. leave your answers in terms of pi.

when the radius doubles, does the area of the circle double?

Radius 2 cm has its area = 4pi square cm
while radius 4 cm has the area = 16 pi square cm...

No,it is not necessary. But, the area becomes four times the area of the previous circle.
 For example, let the radius of a circle be 5 cm.
area= 25pi square cm.
if radius is doubled,i.e., radius becomes 10 , then,
area=100pi square cm.
 
Hence area increases four times...