The dimensions of circle B are twice the dimensions of circle A. The area of circle Bis 64 cm^{2}.What is the area of circle A?

the radii are in the ratio 1:2

so the area has to be in the ratio $$1^2:2^2\quad=\quad 1:4$$

the big one's area is 64pi

so the little ones area is $$64\pi/4 = 16\pi \quad cm^2$$

If the area of circle B = 64*pi cm^2, it's radius must be 8 cm.

Then, the radius of circle A must be 1/2 of that = 4cm

And it's area = pi*(4cm)^2 = 16pi cm^2