AMC 8 · 2006 · #7

Grade 7 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

area-circlesperimeterformula-substitution identify-subproblems ↑ Prerequisites: area-circlesperimeter

📏 Short solution 💡 3 insights

Problem

Three circles are described in three different ways. Circle X has radius π. Circle Y has circumference 8π. Circle Z has area 9π. List the three circles in order from the smallest radius to the largest.

Pick an answer.

(A)

X, Y, Z

(B)

Z, X, Y

(C)

Y, X, Z

(D)

Z, Y, X

(E)

X, Z, Y

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Use a Formula

The three circles are described in three different languages — radius, circumference, area. To compare them, Tool #12 (Change Representation) says: put them all in the same language. Radius is the natural choice because two of the three quantities are already one short formula away from it. Tool #15 (Use a Formula) then does the heavy lifting: C = 2π r gives Y's radius from its circumference, and A = π r² gives Z's radius from its area. Once all three radii are plain numbers, ordering them is just comparing π with 3 and 4.

1STEP 1

Circle X is already given as a radius — record r_X = π and move on.

r_X = π

2STEP 2

Convert Y's circumference with C = 2π r: 2π r_Y = 8π gives r_Y = 4.

2π r_Y = 8π → r_Y =

\frac{8\pi}{2\pi}

= 4

3STEP 3

Convert Z's area with A = π r²: π r_Z² = 9π gives r_Z² = 9, so r_Z = 3.

π r_Z² = 9π → r_Z² = 9 → r_Z = 3

4STEP 4

Compare the radii as numbers: since 3 ＜ π ＜ 4, we get r_Z ＜ r_X ＜ r_Y.

r_Z = 3 ＜ r_X = π ≈ 3.14 ＜ r_Y = 4

5STEP 5

Read the circles off in that radius order to get choice (B).

Z, X, Y → (B)

Answer

Z, X, Y

Cross-check by converting the other way. From r_Y = 4, circumference = 2π(4) = 8π — matches. From r_Z = 3, area = π(3)² = 9π — matches. And π is famously a little over 3 but well under 4, so X sits between Z and Y. The ordering Z, X, Y is consistent in both directions.

💡Key takeaway

When circles are described in different ways — radius, circumference, area — convert every one into the same form first. Once all three are radii, ordering the circles is just ordering three numbers.

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