Competition · AMC preparation · step 4 of 4

AMC 8 · 2012 · #24

Grade 7 geometry-2d

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

area-circlesarea-rectanglescoordinate-geometryspatial-visualization area-differenceidentify-subproblems ↑ Prerequisites: area-circlespythagorean-theoremcoordinate-geometry

📏 Long solution 💡 4 insights 📊 Diagram

Problem

A circle of radius 2 is cut into four congruent arcs (each a quarter-circle). The four arcs are rejoined into a four-pointed "star" whose sides curve inward. Find the ratio of the area of this star to the area of the original circle.

Pick an answer.

(A)

$\frac{4-\pi}{\pi}$

(B)

$\frac{1}{\pi}$

(C)

$\frac{\sqrt{2}}{\pi}$

(D)

$\frac{\pi-1}{\pi}$

(E)

$\frac{3}{\pi}$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The star is an odd, curved shape — we cannot apply any single area formula to it directly. Tool #1 (Draw a Diagram) is the right first move: by sketching the star inside a square of side 4 whose corners coincide with the star's points, we can SEE that the four "bites" taken out of the square are exactly four quarter-circles of radius 2 — which together form one full circle of radius 2. Tool #7 (Identify Subproblems) then turns the problem into two easy pieces we already know: (a) area of a 4 × 4 square, and (b) area of a circle of radius 2. The star is (a) minus (b), and the ratio falls out by simple division.

1STEP 1

Draw the star on axes

Place the star's four points on the axes and box it in the square x,y = ±2: side 4, so its area is 16.

Bounding square side = 4, A_square = 4 × 4 = 16

2STEP 2

Look at the leftover corners

Each of the four square corners (two sides plus one inward arc) is exactly a quarter-disk of radius 2, area π.

Each corner piece = 1/4 π (2)² = π

3STEP 3

Add the four corner pieces

The four quarter-disks reassemble into one full circle of radius 2 — the original circle — total area 4π.

4 × 1/4π(2)² = π(2)² = 4π

Four quarter-circles of the same radius reassemble into one whole circle. This makes the subtraction extra clean.

The four regions between the star and the square drawn around it fit together to make exactly one full circle of radius 2 — the very circle the arcs were cut from.

▸ Why?

Each of those four regions sits in one corner of the square, walled by the two square sides that meet at a right angle there and by the star's inward-curving arc, which makes it exactly a quarter of a disk of radius 2.

▸ Why?

That inward arc is one of the four radius-2 quarter-arcs the original circle was cut into, and its center falls on the square's corner, so the curved wall stays 2 units from that corner all along.

A circle's radius is constantEvery point on a circle sits exactly one radius from the center, so all radii of a circle are equal.

▸ Why?

The two straight walls each run from that corner out to a neighboring point of the star, and those points are the endpoints of the same radius-2 arc, so each straight wall is a radius of it and measures 2.

A circle's radius is constantEvery point on a circle sits exactly one radius from the center, so all radii of a circle are equal.

▸ Why?

The four corner regions are identical, so sliding them to share one center packs them around that point with no gap or overlap, covering one whole disk of radius 2.

▸ Why?

A quarter turn of the whole star-in-square picture lands it back on itself and carries each corner region onto the next, so the four regions are congruent copies.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

Four matching quarter-pieces that fill the way around a single point with no gap or overlap add back up to the one whole disk they form.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

7.G.B.4Identify Subproblems

4STEP 4

Subtract to get the star

Star area = square minus the four corners = 16 - 4π.

A_star = 16 - 4π

5STEP 5

Form the ratio and simplify

Divide by the circle's area 4π and factor 4 from the top: $\frac{16 - 4π}{4π}$ = $\frac{4-π}{π}$ , choice (A).

A_star/A_circle = (16 - 4π)/4π = (4(4 - π))/4π = (4 - π)/π → (A)

Answer

(4-π)/π

Numerically, π ≈ 3.14, so

\frac{4 - π}{π}

≈

\frac{0.86}{3.14}

≈ 0.27. That says the star takes up about 27% of the original circle's area — which matches the picture: the star is clearly much smaller than the original circle (more than half of the circle's area sits outside the star in those four "crescent" gaps). Also, all five answer choices have π in the denominator, which is consistent with our setup (we divided by 4π).

💡Key takeaway

Draw a square around the star: the four "bites" out of the square fit together to make the original circle, so the star is just square minus circle — Grade 7 circle-area reasoning does the rest.

Draw the star on axes
Look at the leftover corners
Add the four corner pieces
Subtract to get the star
Form the ratio and simplify

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Draw the star on axes

Look at the leftover corners

Add the four corner pieces

Subtract to get the star

Form the ratio and simplify

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