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2001 AMC 10 Problem 18

2001 AMC 10 Problem 18AMC 10 Step by Step

Name: Date: October 28, 2026

Quadrilaterals & Polygon Areas · difficulty 3 of 5 · about 3 min · position P16-20

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

A) $50$
B) $52$
C) $54$
D) $56$
E) $58$

Answer

Key insight

The pattern repeats in 3 by 3 blocks. The four corner squares of a block cover 4 of its 9 units of area, so pentagons cover 5 out of 9.

Solution

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.
Use the side of a small square as the unit of length. One block is a $3 \times 3$ square, so its area is $9$ .
Inside one block, there is a small $1 \times 1$ square in each of the four corners. Together they have area $4$ .
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 $9 - 4 = 5$ .
The pentagon fraction is $\frac{5}{9}$ . As a decimal, $5 \div 9 = 0.5555\ldots$ , which is about $55.6\%$ .
The closest choice is $56$ .

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

Why this works

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.

Alternative approach

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 $\frac59$ .

The trap

Guessing from the picture instead of measuring one block, or treating the middle square of a block as a square tile when it is actually cut up among the four pentagons.

Concise solution

The tiling repeats: the picture is made of identical $3 \times 3$ blocks (measured in units of the small square's side), so the fraction covered by pentagons in one block is the fraction for the whole plane.

Take one block, area $3 \cdot 3 = 9$ . It contains four unit squares, one in each corner, with total area $4$ . Everything else in the block belongs to the four pentagons (each pentagon is a unit square along an edge plus a quarter of the central unit square, area $1 + \frac14$ ), so the pentagons cover $9 - 4 = 5$ .

The pentagon share is $\frac{5}{9} = 0.555\ldots$ , i.e. about $55.6\%$ , and the nearest choice is $56$ .

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

Review in the AoPS books

Introduction to Geometry, Chapter 8: Quadrilaterals
Introduction to Geometry, Chapter 16: The More Things Change. . . (§16.1, §16.2, §16.3, §16.4)
Introduction to Geometry, Chapter 4: Perimeter and Area (§4.2, §4.3)
Introduction to Geometry, Chapter 9: Polygons (§9.3)

Techniques

Cut the figure into known shapes (triangles, rectangles, sectors)