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. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: The hexagon is regular with side length 6.; An arc of radius 3 is centered at each of the six vertices.; Each arc, together with the two sides meeting at that vertex, forms a circular sector inside the hexagon.; The shaded region is inside the hexagon but outside every sector.

Unknowns: The area of the shaded region.

Understand

Plan

Primary tool: #16 Change Focus / Count the Complement

Secondary: #7 Identify Subproblems, #1 Draw a Diagram

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.

Execute — Answer: C

#16 Change Focus / Count the Complement 7.G.B.6 Step 1

Reframe as whole minus cut-outs

The shaded area equals the hexagon's area minus the combined area of the six sectors, because the sectors are exactly the parts removed from inside the hexagon.

$$\text{Shaded} = \text{Area}_{\text{hexagon}} - \text{Area}_{\text{6 sectors}}$$

💡 It is easier to measure the full shape and take away the holes than to trace the leftover directly.

#7 Identify Subproblems 8.G.B.7 Step 2

Area of the hexagon

Split the regular hexagon into 6 equilateral triangles of side 6.
Each triangle has area (sqrt(3)/4) times side squared; its height comes from the Pythagorean theorem.
So each triangle is (sqrt(3)/4)(36) = 9 sqrt(3), and six of them give 54 sqrt(3).

$$6 \cdot \frac{\sqrt{3}}{4}\cdot 6^2 = 6 \cdot 9\sqrt{3} = 54\sqrt{3}$$

💡 A regular hexagon is six identical equilateral triangles fanned around its center.

#7 Identify Subproblems 7.G.B.4 Step 3

Area of the six sectors

Each sector sits in a 120-degree interior angle, so it is 120/360 = 1/3 of a full circle of radius 3.
One sector is (1/3)(pi)(3^2) = 3 pi, and the six sectors together are 6 times 3 pi = 18 pi.

$$6 \cdot \frac{120}{360}\cdot \pi \cdot 3^2 = 6 \cdot 3\pi = 18\pi$$

💡 A sector is just a slice of a whole circle, sized by its share of 360 degrees.

#16 Change Focus / Count the Complement 7.NS.A.3 Step 4

Subtract to finish

Subtract the total sector area from the hexagon's area: 54 sqrt(3) minus 18 pi.
This matches answer (C).

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

💡 The leftover shaded area is exactly what remains after the round corners are removed.

[1] #16 7.G.B.6 The shaded area equals the hexagon's area minus the combined area of the six sec

[2] #7 8.G.B.7 Split the regular hexagon into 6 equilateral triangles of side 6. Each triangle

[3] #7 7.G.B.4 Each sector sits in a 120-degree interior angle, so it is 120/360 = 1/3 of a ful

[4] #16 7.NS.A.3 Subtract the total sector area from the hexagon's area: 54 sqrt(3) minus 18 pi.

Review

Reasonableness: 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).

Alternative: Add up the sector angles first: six sectors of 120 degrees total 720 degrees, which is two full circles of radius 3. So the sectors together cover 2(pi)(3^2) = 18 pi in one stroke, then subtract from 54 sqrt(3).

CCSS standards used (min grade 8)

7.G.B.6 Solve real-world problems involving area, surface area, and volume (Framing the shaded region as the hexagon's area minus the removed sectors.)
8.G.B.7 Apply the Pythagorean theorem to determine unknown side lengths in right triangles (Getting the height of each equilateral triangle to compute the hexagon's area 54 sqrt(3).)
7.G.B.4 Know the formulas for area and circumference of a circle (Computing each 120-degree sector as one third of a radius-3 circle.)
7.NS.A.3 Solve real-world problems involving the four operations with rational numbers (Combining the hexagon and sector areas into the final expression.)

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

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Reframe as whole minus cut-outs

Area of the hexagon

Area of the six sectors

Subtract to finish

Review

CCSS standards used (min grade 8)

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