By what percentage does the area of a circle increase when its radius increase

"Percentage" stems from comparing like qualities.  If we start with a circle of radius r and end with a circle of radius R, then the change in area of this circle would be pi*R^2 - pi*r^2, or pi(R^2-r^2).

The ratio of this change to the original area, which was pi*R^2, would then be

       pi(R^2-r^2)        R^2-r^2
    =
            piR^2               R^2

If you want the percentage change, then multiply the above result by 100%.