AMC 10 · 2007 · #13

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid
circular-sectorarea-circlescoordinate-geometry identify-subproblems ↑ Prerequisites: area-circles
📏 Medium solution 💡 3 insights
Problem
Two circles, each of radius 2, are centered at (2,0) and (0,2). Find the area of the region that lies inside both circles at once.

Pick an answer.

(A)
$\pi -2$
(B)
$\frac{\pi}{2}$
(C)
$\frac{\pi \sqrt{3}}{3}$
(D)
$2(\pi -2)$
(E)
$\pi$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The overlap is a curved lens, and the fastest way to see its shape is to draw both circles and mark where they cross (Tool #1). The lens has no simple area formula on its own, so break it into pieces we can measure: each half is a circular sector with a triangle removed (Tool #7 Identify Subproblems). Finally, since the problem is multiple choice, match the computed value against the five options (Tool #3).

1STEP 1

Find where the circles cross

Both circles pass through (0,0) and (2,2) — each point is exactly 2 from both centers, so these are the lens tips.

Crossing points: (0,0) and (2,2)
2STEP 2

Split the lens into two equal segments

By symmetry each half-lens is one circle's segment: the radii from (2,0) to the tips meet at 90°, so half = sector minus triangle.

∠ = 90°, half-lens = sector - triangle
3STEP 3

Measure the sector and the triangle

The sector is a quarter of the circle's 4π, so it is π; the triangle with legs 2 and 2 is 2. Half-lens = π - 2.

sector=1/4(4π)=π, triangle=1/2(2)(2)=2
4STEP 4

Add the two halves

Doubling the two identical halves gives 2(π - 2), choice (D); (A) π - 2 is only one half.

2 (π-2)
Answer
2(π -2)
Estimate numerically: 2(π-2)≈ 2(3.1416-2)=2(1.1416)≈ 2.28. This is a sliver, comfortably smaller than one whole circle's area 4π≈ 12.6, which is what we expect for an overlap where the centers sit a good distance apart (the distance between centers is 2√2≈ 2.83, more than the radius). A positive value a little above 2 is sensible for a thin lens. Choice (A) π-2≈1.14 is exactly half of our answer — the trap for stopping after one segment.
💡Key takeaway

To measure a curved lens where two circles overlap, cut it into pie-slice sectors with the straight triangles trimmed off, then add the leftover slivers.

  • Find where the circles cross
  • Split the lens into two equal segments
  • Measure the sector and the triangle
  • Add the two halves

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