AMC 10 · 2008 · #17

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid
area-rectanglescircular-sectorarea-circles identify-subproblemsphysical-representation ↑ Prerequisites: area-circles
📏 Medium solution 💡 2 insights
Problem
An equilateral triangle has side length 6. Look at every point that lies outside the triangle yet is within 3 units of some point of the triangle. Those points form a band that hugs the outside of the triangle. Find the area of that band.

Pick an answer.

(A)
$\ 36+24\sqrt{3}$
(B)
$\ 54+9\pi$
(C)
$\ 54+18\sqrt{3}+6\pi$
(D)
$\ \left(2\sqrt{3}+3\right)^2\pi$
(E)
$\ 9\left(\sqrt{3}+1\right)^2\pi$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The band has a curved, odd shape, so there is no single formula for it. Tool #7 (Identify Subproblems) says: cut the band into pieces that each have an easy formula. To see those pieces, Tool #1 (Draw a Diagram) sketches the band and shows it is made of flat strips along the sides and rounded wedges at the corners. Tool #17 (Visualize Spatial Relationships) helps read the exact turning angle of each corner wedge. Add the piece areas and the band's area falls out.

1STEP 1

Draw the band and split it into pieces

Shade the band: each straight side gets a flat rectangle, each corner gets a rounded wedge of radius 3.

band = 3 rectangles + 3 corner wedges
2STEP 2

Add the three side rectangles

Each rectangle is 6 long and 3 wide, so 6 × 3 = 18, and the three of them cover 54.

3 × (6 × 3) = 3 × 18 = 54
3STEP 3

Find the angle of one corner wedge

Around a corner, take away the 60° interior angle and the two 90° rectangle gaps from 360°: the wedge keeps 120°.

360° - 60° - 90° - 90° = 120°
4STEP 4

Combine the three wedges into one full circle

The three 120° wedges of radius 3 snap into one whole circle, so together they are π (3)² = 9π.

3 × 120° = 360°, π (3)² = 9π
5STEP 5

Add the pieces for the total area

Rectangles plus wedges gives the band: 54 + 9π, which is choice (B).

54 + 9π → (B)
Answer
54+9π
The area splits cleanly into a plain number 54 from the rectangles and a π-part 9π from the round corners, so the answer should look like 54 + (something)π. Only (B) 54+9π has that exact shape. Choice (A) has no π at all, and (C) carries an extra √(3) that no piece of this band produces, while (D) and (E) are a single π term with no separate whole-number part. Numerically 54 + 9π ≈ 82.3, a sensible size for a 3-wide band around a triangle of side 6.
💡Key takeaway

The outside band is three flat strips (54) plus three corner slices that join into one full circle (9π), so the area is 54 + 9π.

  • Draw the band and split it into pieces
  • Add the three side rectangles
  • Find the angle of one corner wedge
  • Combine the three wedges into one full circle
  • Add the pieces for the total area

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