AMC 10 · 2010 · #16

Grade 8 geometry-2d

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

area-circlescircular-sectorequilateral-trianglearea-triangles symmetry-argument ↑ Prerequisites: area-circles

📏 Long solution 💡 3 insights

Problem

A square with side length 1 and a circle with radius

\frac{\sqrt{3}}{3}

are drawn so they share the same center. Find the area of the region that is inside the circle but outside the square.

Pick an answer.

(A)

$\dfrac{\pi}{3}-1$

(B)

$\dfrac{2\pi}{9}-\dfrac{\sqrt{3}}{3}$

(C)

$\dfrac{\pi}{18}$

(D)

$\dfrac{1}{4}$

(E)

$\dfrac{2\pi}{9}$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The wanted region has a curvy, awkward shape, so the first job is to see it (Tool #1, Draw a Diagram): compare the radius with the square's half-side and half-diagonal to learn exactly how the circle sits over the square. That comparison shows the circle bulges out past the middle of each of the four sides while the four corners of the square stick out of the circle. By symmetry the leftover region is four identical little pieces, so the problem breaks into 'find one piece, then multiply by four' (Tool #7, Identify Subproblems). Each piece is a circular segment, and the clean way to measure a segment is to take a pie-slice (sector) and cut away the triangle inside it (Tool #16, Change Focus / Count the Complement). The one measurement we need — how wide the circle's opening is at each side — comes from a right triangle and the Pythagorean theorem. Seeing that the resulting triangle is equilateral (Tool #17) hands us the 60° angle for free.

1STEP 1

See how the circle sits

The square reaches $0.5$ to a side and $0.707$ to a corner; the radius $0.577$ falls between, so the circle bulges through all four sides.

0.5 ＜ √(3)/3≈ 0.577 ＜ √(2)/2≈ 0.707

2STEP 2

Break into four equal caps

Four-fold symmetry means the region is four identical circular segments, one past each side, so measure one and multiply by four.

Area = 4 × (one circular segment)

3STEP 3

Find the circle's opening at a side

The center, a side's midpoint, and a crossing point make a right triangle, so half the chord is $h=\sqrt{\frac{1}{3}-\frac{1}{4}}$ , that is $\frac{\sqrt{3}}{6}$ .

h = √((1/√(3))² - (1/2)²) = √(1/3 - 1/4) = √(1/12) = 1/2√(3)

4STEP 4

Spot the equilateral triangle

Doubling gives a chord of $\frac{\sqrt{3}}{3}$ , equal to $r$ , so the two radii and the chord are equilateral: the central angle is $60^\circ$ .

2h = 1/√(3) = r → △ is equilateral → central angle = 60°

5STEP 5

Area of the pie slice

A $60^\circ$ slice is $\frac{1}{6}$ of the circle, whose area is $\pi r^2=\frac{\pi}{3}$ , so the sector is $\frac{\pi}{18}$ .

sector = 1/6 π r² = 1/6·π/3 = π/18

6STEP 6

Cut off the triangle, then multiply by four

One segment is $\frac{\pi}{18}-\frac{\sqrt{3}}{12}$ ; times four gives $\frac{2\pi}{9}-\frac{\sqrt{3}}{3}$ , choice (B).

4(π/18 - √(3)/12) = 4π/18 - 4√(3)/12 = 2π/9 - √(3)/3

Answer

2π/9-√(3)/3

A quick sanity check: the circle's area is π/3 ≈ 1.047 and the square's area is 1, so the circle has only about 0.047 more area than the square. The region inside the circle but outside the square must be smaller than that tiny surplus, because the corners of the square (which lie outside the circle) also carry area. Our answer 2π/9 - √(3)/3 ≈ 0.698 - 0.577 = 0.121 is small and positive, consistent with four thin bulges. Choice (E) 2π/9 is just the four sectors without removing the triangles, and (A) π/3-1 ≈ 0.047 is the full circle minus the full square, which wrongly assumes the square sits entirely inside the circle. Both traps confirm (B) is the honest count.

💡Key takeaway

The circle bulges past the middle of each side, so add up four matching slivers: take a sixth-of-the-circle pie slice, cut away its triangle, and multiply by four.

See how the circle sits
Break into four equal caps
Find the circle's opening at a side
Spot the equilateral triangle
Area of the pie slice
Cut off the triangle, then multiply by four