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AMC 8 · 2006 · #7

Grade 7 geometry-2d
Archetype Arithmetic Expression Decomposition
area-circlesperimeterformula-substitution identify-subproblems ↑ Prerequisites: area-circlesperimeter
📏 Short solution 💡 3 insights

Problem

Circle XX has a radius of π\pi. Circle YY has a circumference of 8π8 \pi. Circle ZZ has an area of 9π9 \pi. List the circles in order from smallest to the largest radius.

Pick an answer.

(A)
X, Y, Z
(B)
Z, X, Y
(C)
Y, X, Z
(D)
Z, Y, X
(E)
X, Z, Y
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Toolkit + CCSS Solution

Understand

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

Givens: Circle $X$: radius $= \pi$; Circle $Y$: circumference $= 8\pi$; Circle $Z$: area $= 9\pi$; Answer choices: (A) $X, Y, Z$; (B) $Z, X, Y$; (C) $Y, X, Z$; (D) $Z, Y, X$; (E) $X, Z, Y$

Unknowns: The ordering of $X, Y, Z$ by radius, smallest to largest

Understand

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

Givens: Circle $X$: radius $= \pi$; Circle $Y$: circumference $= 8\pi$; Circle $Z$: area $= 9\pi$; Answer choices: (A) $X, Y, Z$; (B) $Z, X, Y$; (C) $Y, X, Z$; (D) $Z, Y, X$; (E) $X, Z, Y$

Plan

Primary tool: #15 Use a Formula

Secondary: #12 Change Representation

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: $C = 2\pi r$ gives $Y$'s radius from its circumference, and $A = \pi r^2$ gives $Z$'s radius from its area. Once all three radii are plain numbers, ordering them is just comparing $\pi$ with $3$ and $4$.

Execute — Answer: B

#12 Change Representation 7.G.B.4 Step 1
  • Circle $X$ is already given in radius form.
  • Record it.
$$r_X = \pi$$

💡 Nothing to convert — $X$ already speaks the radius language.

#15 Use a Formula 7.G.B.4 Step 2
  • Translate circle $Y$ from circumference to radius using $C = 2\pi r$.
  • Set $2\pi r_Y = 8\pi$ and solve.
$$2\pi r_Y = 8\pi \;\Rightarrow\; r_Y = \dfrac{8\pi}{2\pi} = 4$$

💡 The Grade 7 circumference formula is the bridge from $C$ to $r$. Dividing by $2\pi$ is a one-step equation.

#15 Use a Formula 7.G.B.4 Step 3
  • Translate circle $Z$ from area to radius using $A = \pi r^2$.
  • Set $\pi r_Z^2 = 9\pi$ and solve.
$$\pi r_Z^2 = 9\pi \;\Rightarrow\; r_Z^2 = 9 \;\Rightarrow\; r_Z = 3$$

💡 The Grade 7 area formula gives $r^2 = 9$, and the positive square root is $3$.

#12 Change Representation 7.NS.A.3 Step 4
  • Now compare the three radii as plain numbers.
  • Use the approximation $\pi \approx 3.14$, so $3 < \pi < 4$.
$$r_Z = 3 \;<\; r_X = \pi \approx 3.14 \;<\; r_Y = 4$$

💡 Once every circle is described by the same quantity, ordering them is just ordering three numbers on the number line.

#15 Use a Formula 7.G.B.4 Step 5

Read off the order from smallest to largest radius.

$$Z, X, Y \;\Rightarrow\; \textbf{(B)}$$

💡 The ordering of the radii is the ordering of the circles.

[1] #12 7.G.B.4 Circle $X$ is already given in radius form. Record it.
[2] #15 7.G.B.4 Translate circle $Y$ from circumference to radius using $C = 2\pi r$. Set $2\pi
[3] #15 7.G.B.4 Translate circle $Z$ from area to radius using $A = \pi r^2$. Set $\pi r_Z^2 = 9
[4] #12 7.NS.A.3 Now compare the three radii as plain numbers. Use the approximation $\pi \approx
[5] #15 7.G.B.4 Read off the order from smallest to largest radius.

Review

Reasonableness: Cross-check by converting the other way. From $r_Y = 4$, circumference $= 2\pi(4) = 8\pi$ — matches. From $r_Z = 3$, area $= \pi(3)^2 = 9\pi$ — matches. And $\pi$ 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.

Alternative: Tool #6 (Guess & Check on the answer choices): only options (B) and (E) place $X$ in the middle or end. Since $r_X = \pi$ is clearly between $3$ and $4$, $X$ cannot be the largest, ruling out (E). That leaves (B) directly, without computing $r_Y$ and $r_Z$ from scratch.

CCSS standards used (min grade 7)

  • 7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems (Applying $C = 2\pi r$ to recover $r_Y = 4$ from circumference $8\pi$, and $A = \pi r^2$ to recover $r_Z = 3$ from area $9\pi$.)
  • 7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers (Ordering the three radii $3$, $\pi$, and $4$ on the number line using $\pi \approx 3.14$.)

⭐ 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.

⭐ 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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