AMC 10 · 2011 · #11

Grade 8 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid

area-circlestangent-circlescoordinate-geometry identify-subproblemscomplementary-counting ↑ Prerequisites: area-circles

📏 Long solution 💡 3 insights

Problem

Two equal circles touch, and a third rests on the midpoint between their centres. Find the part of the third outside both.

Pick an answer.

(A)

$3 - \frac{\pi}{2}$

(B)

$\frac{\pi}{2}$

(C)

(D)

$\frac{3\pi}{4}$

(E)

$1+\frac{\pi}{2}$

How to solve

Strategy Identify Subproblems

The wanted region is an awkward curved shape, but it is really the whole disk C with two overlapping bites removed, so Tool #7 (Identify Subproblems) is the anchor: compute Area(C), then the overlap C∩ A, then double it by symmetry. Tool #1 (Draw a Diagram) makes this exact — placing the circles on a coordinate grid fixes every center and every crossing point. Tool #17 (Visualize Spatial Relationships) is what reveals that each overlap lens is built from two congruent quarter-circle-minus-triangle segments, which is the move that makes the messy π terms cancel.

1STEP 1

Place the circles on a grid

Coordinates make every centre explicit.

A=(-1,0), B=(1,0), M=(0,0), C=(0,1); AC=√(1²+1²)=√(2)

2STEP 2

Break the target into a subtraction

The target is the whole circle minus two lenses.

Area=π-2·Area(C∩ A)

3STEP 3

Find where C and A cross

The crossing points are easy to read off.

C∩ A={(0,0), (-1,1)}

4STEP 4

The arc spans a right angle

The arc spans a right angle.

∠ MAP=90° → sector=90/360π(1)²=π/4

5STEP 5

Peel the triangle to get the lens

A lens is a sector minus a triangle.

△ AMP=1/2(1)(1)=1/2; Area(C∩ A)=2(π/4-1/2)=π/2-1

6STEP 6

Subtract both overlaps

The circle constant cancels, leaving 2, choice (C).

π-2(π/2-1)=π-π+2=2 → (C)

Answer

Circle C has area π≈3.14. Each overlap is π/2-1≈0.57, and two of them total about 1.14, leaving roughly 3.14-1.14=2.0 — matching the exact answer 2. It is also reassuring that every π term cancels: the leftover region can be reshaped into a straight-sided figure of whole-number area, so a π-free answer like 2 is exactly what to expect. The π-laden options (B) π/2, (D) 3π/4, and (E) 1+π/2 are traps for stopping before the cancellation.

💡Key takeaway

To find a leftover area, take the whole shape and subtract each overlapping bite — and a quarter-circle minus its triangle turns the messy π pieces into a clean number.

Place the circles on a grid
Break the target into a subtraction
Find where C and A cross
The arc spans a right angle
Peel the triangle to get the lens
Subtract both overlaps