AMC 10 · 2014 · #12

Grade 8 geometry-2d

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

area-regular-hexagoncircular-sectorarea-difference complementary-countingidentify-subproblems ↑ Prerequisites: area-regular-hexagonarea-circles

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

A regular hexagon has side length 6. At each of its six vertices, a circular arc of radius 3 is drawn, cutting a small circular sector out of that corner of the hexagon. Find the area of the part that is inside the hexagon but outside all six sectors.

Pick an answer.

(A)

$27\sqrt{3}-9\pi$

(B)

$27\sqrt{3}-6\pi$

(C)

$54\sqrt{3}-18\pi$

(D)

$54\sqrt{3}-12\pi$

(E)

$108\sqrt{3}-9\pi$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

The shaded region has a jagged, hard-to-measure boundary. Instead of measuring it directly, take the whole hexagon and subtract the pieces that were removed. That turns one messy region into two clean pieces: the hexagon's area and the total sector area.

1STEP 1

Reframe as whole minus cut-outs

The sectors are exactly what was removed from inside, so shaded area is the hexagon's area minus the six sectors.

Shaded = Area_hexagon - Area₆ sectors

2STEP 2

Area of the hexagon

Split it into 6 equilateral triangles of side 6; each has area $9\sqrt{3}$ , so the hexagon is $54\sqrt{3}$ .

6 · √(3)/4 · 6² = 6 · 9√(3) = 54√(3)

3STEP 3

Area of the six sectors

Each sector fills a 120-degree interior angle, so it is $\frac{1}{3}$ of a radius-3 circle: $3\pi$ each, $18\pi$ in all.

6 · 120/360 · π · 3² = 6 · 3π = 18π

4STEP 4

Subtract to finish

Take $18\pi$ off $54\sqrt{3}$ , leaving $54\sqrt{3}-18\pi$ — choice (C).

54√(3) - 18π

Answer

54√(3)-18π

Numerically

54\sqrt{3}

is about 93.5 and

18\pi

is about 56.5, so the shaded area is about 37. That is positive and smaller than the hexagon's area, which is exactly what a region-with-holes should be, and it matches choice (C).

💡Key takeaway

To find a leftover region with rounded bites taken out, measure the whole shape and subtract the pieces that were cut away.

Reframe as whole minus cut-outs
Area of the hexagon
Area of the six sectors
Subtract to finish

Reframe as whole minus cut-outs

Area of the hexagon

Area of the six sectors

Subtract to finish

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