AMC 10 · 2016 · #15

Grade 8 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference
area-circlestangent-circles area-difference ↑ Prerequisites: area-circles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
Seven circular cookies of radius 1 are cut from one big circle of dough. One cookie sits at the center, six surround it; neighbors touch, and every outer cookie also touches the edge of the dough. All the leftover dough is rolled into one more cookie of the same thickness. Find that scrap cookie's radius.

Pick an answer.

(A)
$\sqrt{2}$
(B)
1.5
(C)
$\sqrt{\pi}$
(D)
$\sqrt{2\pi}$
(E)
$\pi$

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The picture controls everything, so tool #1 (Draw a Diagram) reads off the one missing length: the big dough circle's radius, found by walking straight out from the center along a row of touching cookies. From there the question is pure bookkeeping of areas, so tool #7 (Identify Subproblems) splits it into three small jobs: the big circle's area, the seven small circles' area, and the leftover that becomes the scrap cookie. Equal thickness means equal area, so the scrap's radius drops out of π r² = leftover area.

1STEP 1

Find the big circle's radius

Walk straight out from the center across one cookie's radius plus an outer cookie's diameter, so the dough radius is 3.

R = 1 + 2 = 3
2STEP 2

Area of dough and of one cookie

By π r², the radius-3 dough has area 9π and each radius-1 cookie has area π.

dough = π · 3² = 9π, one cookie = π · 1² = π
3STEP 3

Leftover dough area

The seven cookies use 7π of dough, so the scrap left over is 9π − 7π = 2π.

scrap area = 9π - 7 · π = 9π - 7π = 2π
4STEP 4

Turn the area into a radius

Set π r² = 2π, cancel the π, and take the square root of r² = 2 to get the radius.

π r² = 2π → r² = 2 → r = √(2), which is choice (A)
Answer
√(2)
Check the area splits cleanly: 9π of dough, 7π in cookies, 2π in scrap, and 7π + 2π = 9π, so no dough is lost or invented. The scrap radius √(2)≈ 1.41 sits sensibly between a single 1-radius cookie and the 3-radius dough, and √(2) is exactly one of the choices, confirming (A).
💡Key takeaway

The dough is 9π, the seven cookies use 7π, so the 2π scrap makes a cookie with r²=2, meaning r=√(2).

  • Find the big circle's radius
  • Area of dough and of one cookie
  • Leftover dough area
  • Turn the area into a radius

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