Competition · AMC preparation · step 4 of 4

AMC 8 · 2000 · #19

Grade 7 geometry-2d

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

area-circlesarea-trianglesspatial-visualizationreflection-symmetry area-differenceidentify-subproblems ↑ Prerequisites: area-circlesarea-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A region is bounded by three circular arcs, each of radius 5. Two of them, arcs AB and AD, are quarter-circles that bulge inward at the bottom corners; the third, arc BCD, is a semicircle that bulges outward at the top. Find the area of the region.

Pick an answer.

(A)

(B)

$10+5\pi$

(C)

(D)

$50+5\pi$

(E)

$25\pi$

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The figure is a curvy blob, but the asy coordinates pin down a clean rectangle in the middle: B=(-5,5) and D=(5,5) sit directly above A=(0,0). Tool #1 (Draw a Diagram) makes the trick visible — cut the region with the segment BD. Above BD is exactly the semicircle on diameter BD. Below BD is a 10 × 5 rectangle with two quarter-circles scooped out of its bottom corners. Tool #7 (Break Into Subproblems) then turns the area into three easy circle/rectangle areas. The payoff: the semicircle added on top has the same area as the two quarter-circles scooped out of the bottom (25π/2 = 2 · 25π/4), so all the π terms cancel and only the rectangle is left.

1STEP 1

Read the coordinates off the figure

Draw segment BD. From B=(-5,5) and D=(5,5) it is horizontal with length 10, splitting the region into a top and a bottom piece.

BD = 5 - (-5) = 10

2STEP 2

Find the upper area

Top piece: arc BCD is a semicircle of diameter 10, radius 5, so its area is 25π/2.

A_upper = 1/2π(5)² = 25π/2

3STEP 3

Find the lower area

Bottom piece: a 10×5 rectangle (area 50) with a radius-5 quarter-circle scooped from each bottom corner, so its area is 50 - 25π/2.

A_lower = 10 × 5 - 2 · 1/4π(5)² = 50 - 25π/2

4STEP 4

Add the two pieces

Add them: the + $\frac{25π}{2}$ on top and the - $\frac{25π}{2}$ below cancel, leaving just the rectangle — area 50.

A_total = 25π/2 + (50 - 25π/2) = 50 → (C)

Combining like terms: the π pieces are opposites and add to zero, leaving the plain number 50.

The whole region has the same area as the 10 × 5 rectangle, 50 square units, because the semicircle added on top has exactly the same area as the two quarter-circles scooped from the bottom, so they cancel.

▸ Why?

Cutting along segment BD breaks the region, with no gaps and no overlaps, into the semicircle on top and the rectangle-with-two-scoops on the bottom, so the region's area is the rectangle's area minus the two scoops plus the semicircle.

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

▸ Why?

The semicircle on top and the two scooped quarter-circles have equal area, so 'minus the two scoops plus the semicircle' adds nothing overall and the region's area equals the rectangle's area alone.

▸ Why?

Each scooped quarter-circle slides up without stretching to fill exactly one half of the semicircle, and moving a shape does not change its area, so the two quarter-circles and the semicircle have equal area.

▸ Why?

Sliding or turning a shape lays it exactly onto a copy, so every length and angle stays the same and its area does not change.

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?

Each scoop and each half of the semicircle is a quarter-turn wedge reaching one radius out from a corner, and all these radii equal 5, so the wedges match in shape and size.

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?

Removing an area and then adding the same area back returns you to where you started, since the two operations undo each other.

Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.

▸ Why?

The rectangle is 10 wide and 5 tall, and its area is 5 rows of 10 unit squares, which is 10 times 5.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

6.EE.A.3Identify Subproblems

Answer

The two quarter-circles cut out at the bottom together form half of a radius-5 circle, the same shape as the semicircle glued on top. Mentally pick up each scooped quarter and slide it up to fill its matching half of the semicircle: the curvy region rearranges into the 10 × 5 rectangle exactly, area 50. That rules out (A) 25 (too small — even the rectangle alone is 50), (E) 25π ≈ 78.5 (too big — the region fits inside the 10 × 10 bounding square of area 100 but is clearly less than the bounding square), and (B), (D), which would mean the π terms do not cancel. The clean integer answer 50 matches (C).

💡Key takeaway

Cut the region with segment BD — the semicircle glued on top has the same area as the two quarter-circles scooped out of the rectangle below, so the π terms cancel and the answer is just the rectangle, 10 × 5 = 50.

Read the coordinates off the figure
Find the upper area
Find the lower area
Add the two pieces

A parent dashboard for the family lives at sensimlab.com.

Read the coordinates off the figure

Find the upper area

Find the lower area

Add the two pieces

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