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AMC 8 · 2014 · #20

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference
area-rectanglesarea-circlesestimation area-differenceidentify-subproblems ↑ Prerequisites: area-rectanglesarea-circles
📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Rectangle ABCDABCD has sides CD=3CD=3 and DA=5DA=5. A circle of radius 11 is centered at AA, a circle of radius 22 is centered at BB, and a circle of radius 33 is centered at CC. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

(A) 3.5(B) 4.0(C) 4.5(D) 5.0(E) 5.5\text{(A) }3.5\qquad\text{(B) }4.0\qquad\text{(C) }4.5\qquad\text{(D) }5.0\qquad\text{(E) }5.5

Pick an answer.

(A)
3.5
(B)
4.0
(C)
4.5
(D)
5.0
(E)
5.5
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Toolkit + CCSS Solution

Understand

Restated: A $3 \times 5$ rectangle $ABCD$ has three circles centered at three of its corners: radius $1$ at $A$, radius $2$ at $B$, radius $3$ at $C$. Only the parts of the circles that fall inside the rectangle matter. Find the area of the rectangle that is not covered by any circle, then pick the closest answer choice.

Givens: Rectangle sides: $CD = 3$ and $DA = 5$; Circle at $A$ has radius $1$; Circle at $B$ has radius $2$ (and $AB = CD = 3$); Circle at $C$ has radius $3$ (and $BC = DA = 5$); Answer choices: (A) $3.5$, (B) $4.0$, (C) $4.5$, (D) $5.0$, (E) $5.5$

Unknowns: The area of the region inside the rectangle but outside all three circles, rounded to the nearest choice

Understand

Restated: A $3 \times 5$ rectangle $ABCD$ has three circles centered at three of its corners: radius $1$ at $A$, radius $2$ at $B$, radius $3$ at $C$. Only the parts of the circles that fall inside the rectangle matter. Find the area of the rectangle that is not covered by any circle, then pick the closest answer choice.

Givens: Rectangle sides: $CD = 3$ and $DA = 5$; Circle at $A$ has radius $1$; Circle at $B$ has radius $2$ (and $AB = CD = 3$); Circle at $C$ has radius $3$ (and $BC = DA = 5$); Answer choices: (A) $3.5$, (B) $4.0$, (C) $4.5$, (D) $5.0$, (E) $5.5$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #1 Draw a Diagram

The shaded region is the rectangle with three corner pieces removed, so the natural move is Tool #7 (Subproblems): compute the rectangle's area, compute each quarter-circle's area, then subtract. Tool #1 (Diagram) keeps us honest about *which* part of each circle sits inside the rectangle — at a corner with a $90^\circ$ angle, exactly one quarter does — and lets us check on the picture that the quarter-circles do not overlap before we add them up.

Execute — Answer: B

#7 Identify Subproblems 4.MD.A.3 Step 1

Compute the rectangle's area from its side lengths $DA = 5$ and $CD = 3$.

$$\text{Area}_{\text{rect}} = 5 \times 3 = 15$$

💡 Length times width for a rectangle is the Grade 4 area formula — this is the "whole" we will carve corners out of.

#1 Draw a Diagram 7.G.B.4 Step 2
  • At each rectangle corner, the interior angle is $90^\circ$, so the part of a circle centered there that lies inside the rectangle is a quarter of the full disk.
  • Each quarter-circle area is $\tfrac{1}{4}\pi r^2$.
$$\text{Area}_A = \tfrac{1}{4}\pi(1)^2 = 0.25\pi, \quad \text{Area}_B = \tfrac{1}{4}\pi(2)^2 = \pi, \quad \text{Area}_C = \tfrac{1}{4}\pi(3)^2 = 2.25\pi$$

💡 A $90^\circ$ corner sweeps exactly $\tfrac{90}{360} = \tfrac{1}{4}$ of the disk into the rectangle — applying the Grade 7 circle-area formula.

#7 Identify Subproblems 7.G.B.6 Step 3
  • Check that the three quarter-circles do not overlap each other.
  • On side $AB$ (length $3$), the $A$-radius $1$ plus the $B$-radius $2$ equals $3$, so they just touch.
  • On side $BC$ (length $5$), the $B$-radius $2$ plus the $C$-radius $3$ equals $5$, so they also just touch.
  • Circles at $A$ and $C$ are even farther apart (diagonal).
  • So the quarter-circles meet only at single points and their areas can be added with no double-counting.
$$1 + 2 = 3 = AB, \quad 2 + 3 = 5 = BC$$

💡 Splitting the corner pieces into separate disjoint subproblems is only legal if they don't overlap — the side-length check confirms it.

#7 Identify Subproblems 7.G.B.6 Step 4

Add the three disjoint quarter-circle areas to get the total covered area inside the rectangle.

$$\text{Total covered} = 0.25\pi + \pi + 2.25\pi = 3.5\pi$$

💡 Disjoint pieces add — the heart of the Subproblems tool.

#7 Identify Subproblems 7.G.B.4 Step 5

Subtract the covered area from the rectangle, then plug in $\pi \approx 3.14$ and compare to the answer choices.

$$15 - 3.5\pi \approx 15 - 3.5(3.14) = 15 - 10.99 = 4.01 \;\Rightarrow\; \textbf{(B) } 4.0$$

💡 Whole minus covered = uncovered. The numerical value $4.01$ is closest to choice (B) $4.0$.

[1] #7 4.MD.A.3 Compute the rectangle's area from its side lengths $DA = 5$ and $CD = 3$.
[2] #1 7.G.B.4 At each rectangle corner, the interior angle is $90^\circ$, so the part of a cir
[3] #7 7.G.B.6 Check that the three quarter-circles do not overlap each other. On side $AB$ (le
[4] #7 7.G.B.6 Add the three disjoint quarter-circle areas to get the total covered area inside
[5] #7 7.G.B.4 Subtract the covered area from the rectangle, then plug in $\pi \approx 3.14$ an

Review

Reasonableness: Sanity-check the magnitude. The rectangle has area $15$. The three quarter-circles together are $3.5\pi \approx 11$, which is a bit more than two-thirds of the rectangle — visually that matches the figure, where the big circle at $C$ alone eats up a large corner. So the leftover should be a small single-digit number, and $15 - 11 = 4$ lands right in the middle of the choice range $(3.5, 4.0, 4.5, 5.0, 5.5)$. Choice (B) is the only one consistent with $3.5\pi$ very close to $11$.

Alternative: Tool #16 (Change Focus / Complement) frames the same calculation as "don't compute the uncovered area directly; compute the covered area and subtract from the whole" — which is exactly what we did. Tool #3 (Eliminate) on the choices also works: the answer equals $15 - 3.5\pi$, and trying $\pi \approx 3.14$ gives $4.01$, which only matches (B); choices (A) $3.5$ and (C) $4.5$ would need $\pi$ values far from $3.14$.

CCSS standards used (min grade 7)

  • 4.MD.A.3 Apply the area and perimeter formulas for rectangles in real-world and mathematical problems (Computing the rectangle's area as $5 \times 3 = 15$.)
  • 7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems (Computing each quarter-circle area $\tfrac{1}{4}\pi r^2$ for $r = 1, 2, 3$, and approximating $3.5\pi$ with $\pi \approx 3.14$ at the end.)
  • 7.G.B.6 Solve real-world and mathematical problems involving area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms (Verifying the three quarter-circles are disjoint (radii sum to the side lengths) and combining their areas additively, then subtracting from the rectangle to get the uncovered region.)

⭐ This AMC 8 problem only needs Grade 7 circle-area know-how plus the "break the shape into pieces and subtract" idea you already use for L-shaped figures.

⭐ This AMC 8 problem only needs Grade 7 circle-area know-how plus the "break the shape into pieces and subtract" idea you already use for L-shaped figures.

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