AMC 10 · 2004 · #9

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference └ Sub-archetype Circle-Polygon Hybrid
area-circlesarea-rectanglescircular-sector physical-representationidentify-subproblems ↑ Prerequisites: area-circles
📏 Medium solution 💡 2 insights
Problem
A square has side length 10. A circle of radius 10 is centered at one corner of that square. Find the area of the union of the two regions — every point that lies inside the square, the circle, or both.

Pick an answer.

(A)
$200+25\pi$
(B)
$100+75\pi$
(C)
$75+100\pi$
(D)
$100+100\pi$
(E)
$100+125\pi$

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

A picture is the whole game here (#1 Draw a Diagram): once you place the circle on a corner you can see that only a corner wedge of the circle sits inside the square. Then the union splits into three clean pieces to compute — square area, circle area, and the overlap — which is Identify Subproblems (#7), combined by the union rule so the shared part is not double-counted. Finally Eliminate Possibilities (#3) checks the result against the five choices, since the traps come from mishandling the overlap.

1STEP 1

Draw it and spot the overlap

Center the circle on a corner: the two sides meeting there open a right angle, so exactly a quarter of the disk lands inside the square.

corner angle=90°=1/4 of a full turn
2STEP 2

Area of the square and the circle

Side times side, then π times radius squared: 100 and 100π, kept separate until the shared part is handled.

square=10 · 10=100, circle=π · 10²=100π
3STEP 3

Measure the shared corner wedge

That quarter-disk fits entirely inside the 10 by 10 square, so the shared area is one fourth of the circle: 25π.

overlap=1/4 · 100π=25π
4STEP 4

Combine without double-counting

Add 100 and 100π, then subtract the twice-counted 25π once: the union is 100+75π, choice (B).

100+100π-25π=100+75π → (B)
Answer
100+75π
The answer should be a bit less than the naive sum 100+100π, because the square and circle overlap; 100+75π is exactly 25π smaller, which matches the quarter-circle they share. It should also be more than the circle alone (100π) minus that quarter — indeed 100+75π≈ 100+235.6=335.6, larger than the square (100) and larger than three-quarters of the circle (75π≈235.6), as a union of the two must be. Choice (D) ignores the overlap; (A), (C), (E) use the wrong overlap fraction. Only (B) fits.
💡Key takeaway

When two shapes overlap, add both areas and subtract the shared piece once — here the square grabs only a quarter of the circle, so the union is 100+100π-25π=100+75π.

  • Draw it and spot the overlap
  • Area of the square and the circle
  • Measure the shared corner wedge
  • Combine without double-counting

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