AMC 10 · 2006 · #6

Grade 7 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

circle-circumferenceperimeterspatial-visualization identify-subproblems ↑ Prerequisites: circle-circumferenceperimeter

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\frac{2}{\pi}$ , as shown. What is the perimeter of this region?

Pick an answer.

(A)

$\frac{4}{\pi}$

(B)

(C)

$\frac{8}{\pi}$

(D)

(E)

$\frac{16}{\pi}$

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: A square has sides of length 2/π. On each side, a semicircular arc is built using that side as its diameter, and the four arcs bulge outward to bound a flower-like region. Find the perimeter of that region.

Givens: A square whose sides each measure 2/π; A semicircular arc is built on each of the four sides; Each side of the square is the diameter of its semicircle; The four arcs together bound one region

Unknowns: The perimeter of the bounded region

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #1 Draw a Diagram, #17 Visualize Spatial Relationships

The perimeter is one curvy loop, but it is built from four identical semicircular arcs. Break the problem into one small piece: find the length of a single arc, then multiply by four. Finding one arc needs the circle circumference formula, which first needs the radius from the given side.

Execute — Answer: D

#17 Visualize Spatial Relationships 7.G.B.4 Step 1

Side is the diameter

Each semicircle uses a full side of the square as its diameter.
The side measures 2/π, so the diameter is 2/π.
The radius is half of the diameter, so the radius is (1/2)·(2/π) = 1/π.

$$r = \frac{1}{2}\cdot\frac{2}{\pi} = \frac{1}{\pi}$$

💡 The radius is always half the diameter, and here the diameter is just the square's side.

#7 Identify Subproblems 7.G.B.4 Step 2

Length of one arc

A full circle of radius r has circumference 2πr.
A semicircular arc is half of that, so its length is πr.
Put in r = 1/π: the arc length is π·(1/π) = 1.
The π on top cancels the π on the bottom, leaving 1.

$$\text{arc} = \pi r = \pi\cdot\frac{1}{\pi} = 1$$

💡 Half a circle's edge is π times the radius, and the chosen radius makes the π cancel cleanly.

#7 Identify Subproblems 3.OA.C.7 Step 3

Add the four arcs

The region's boundary is exactly four of these semicircular arcs, one per side of the square.
Each has length 1, so the perimeter is 4·1 = 4.
The answer is (D).

$$P = 4\cdot 1 = 4$$

💡 Four equal arcs make a perimeter four times the length of one arc.

[1] #17 7.G.B.4 Each semicircle uses a full side of the square as its diameter. The side measure

[2] #7 7.G.B.4 A full circle of radius r has circumference 2πr. A semicircular arc is half of t

[3] #7 3.OA.C.7 The region's boundary is exactly four of these semicircular arcs, one per side o

Review

Reasonableness: The result 4 is one of the listed choices and is a clean whole number, which fits how the side 2/π was chosen so the π cancels. As a sanity check, four semicircles equal two full circles; two circumferences are 2·(2πr) = 4πr = 4π·(1/π) = 4, the same answer. The straight sides were correctly ignored because they lie inside the region.

Alternative: Combine the arcs first instead of last: four semicircles join into two whole circles, each with circumference 2πr = 2π·(1/π) = 2. Two of them give 2·2 = 4, the same perimeter.

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 (Getting the radius from the diameter and using the circumference formula to find each semicircular arc length)
3.OA.C.7 Fluently multiply and divide within 100 (Multiplying the four equal arc lengths to get the total perimeter)

⭐ When a boundary is made of identical curved pieces, measure one piece and multiply, and a cleverly chosen radius can make the π cancel to a whole number.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Side is the diameter

Length of one arc

Add the four arcs

Review

CCSS standards used (min grade 7)

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