AMC 10 · 2009 · #6

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference
area-circlesarea-differencefraction-arithmetic identify-subproblemscomplementary-counting ↑ Prerequisites: area-circles
📏 Medium solution 💡 2 insights 📊 Diagram
Problem
A circle of radius 2 sits snugly inside a semicircle, touching the flat straight edge and the curved arc. The region inside the semicircle but outside the circle is shaded. Find what fraction of the semicircle's area is shaded.

Pick an answer.

(A)
$\ \frac{1}{2}$
(B)
$\ \frac{\pi}{6}$
(C)
$\ \frac{2}{\pi}$
(D)
$\ \frac{2}{3}$
(E)
$\ \frac{3}{\pi}$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The one fact the picture hides is the semicircle's radius. Reading the figure carefully (Tool #1) unlocks it: the circle stands on the flat edge and its top just reaches the arc, so its full diameter fits between them and equals the semicircle's radius. Once both radii are known, the area question splits into two clean pieces (Tool #7): the semicircle's area and the circle's area. The shaded part is not computed directly but as what is left over after removing the circle (Tool #16), which turns the fraction into a simple subtraction and cancellation.

1STEP 1

Read the semicircle's radius from the figure

The center sits 2 above the flat edge and the top 2 more, so the semicircle's radius is 4.

R_semi = 2 + 2 = 4
2STEP 2

Area of the semicircle

A semicircle is half a full circle: radius 4 gives π·4² = 16π, so half of that is 8π.

A_semi = 1/2π (4)² = 1/2 · 16π = 8π
3STEP 3

Area of the inscribed circle

The inscribed circle has radius 2, so the same formula gives area 4π.

A_circ = π (2)² = 4π
4STEP 4

Subtract, then form the fraction

Shaded is what's left after removing the circle: 8π - 4π = 4π, and 4π over 8π reduces to 1/2.

A_shaded/A_semi = (8π - 4π)/8π = 4π/8π = 1/2 (A)
Answer
1/2
The circle's area 4π is exactly half the semicircle's area 8π, so the shaded leftover must also be 4π — the two pieces split the semicircle evenly. A fraction of 1/2 is one of the listed choices and matches the picture, where the white circle visibly fills about half the region. Note every π cancels, so the answer is a clean rational number, which fits choice (A) rather than the π-containing options (B), (C), (E).
💡Key takeaway

The circle's diameter equals the semicircle's radius, so the circle covers exactly half the semicircle and the shaded leftover is the other half, 1/2.

  • Read the semicircle's radius from the figure
  • Area of the semicircle
  • Area of the inscribed circle
  • Subtract, then form the fraction

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