let

d = the diameter of the circle, d>0

r = d/2 the radius of the circle, r>0

A = (r^2) pi is the area of the circle with radius r

the area is quadrupled when the length of its radius is increased by 4 cm:

4A = (r + 4)^2 pi

4 (r^2) pi = (r + 4)^2 pi divide both sides by pi

4 (r^2) = (r + 4)^2

by solving we find:

r = 4 cm

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d = 2*4

d = 8 cm

the diameter of the circle is 8 cm.