2001 AMC 10 Problem 18

1. The plane goes on forever, so we cannot measure all of it. But the picture is one block repeated over and over, so the fraction covered by pentagons in one block is the fraction for the whole plane.
2. Use the side of a small square as the unit of length. One block is a $3 \times 3$ square, so its area is $9$.
3. Inside one block, there is a small $1 \times 1$ square in each of the four corners. Together they have area $4$.
4. Everything else in the block is pentagon. (Each pentagon is the $1 \times 1$ square in the middle of one edge, plus a triangle that is one quarter of the center square.) So the pentagons cover $9 - 4 = 5$.
5. The pentagon fraction is $\frac{5}{9}$. As a decimal, $5 \div 9 = 0.5555\ldots$, which is about $55.6\%$.
6. The closest choice is $56$.

The answer is $\boxed{\textbf{(D)}\ 56}$.

For a pattern that repeats forever, "what fraction of the plane" is the same as "what fraction of one repeating block." Pick the block, then count the easy pieces (here, the squares) and subtract, rather than working out the area of the odd-shaped pieces directly. The answer choices are only $2$ apart, so get an exact fraction before rounding.

Find the area of one pentagon directly. It is a $1 \times 1$ square (area $1$) plus a quarter of the center square (area $\frac14$), so its area is $\frac54$. Four pentagons give $4 \times \frac54 = 5$ out of a block of area $9$, the same $\frac59$.