jmc

Juan can measure the diameter as a length anywhere between $20-20\cdot 0.2=16$ and $20+20\cdot 0.2=24$ cm. The actual area of the circle is $\pi (20/2{)}^2=100\pi$ square cm, but Juan can compute the area anywhere in the range $\pi (16/2{)}^2=64\pi$ square cm to $\pi (24/2{)}^2=144\pi$ square cm. Using the lower bound of the range, Juan's error is $(100\pi -64\pi )/(100\pi )=36\%$. Using the upper bound of the range, Juan's error is $(144\pi -100\pi )/(100\pi )=44\%$. Thus, the largest possible percent error is $\boxed{44}$ percent.