AMC 10 · 2005 · #8
Grade 7 geometry-2dAn 8-foot by 10-foot bathroom floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
Pick an answer.
AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A bathroom floor measures $8$ feet by $10$ feet and is covered by $1$-foot-by-$1$-foot square tiles. On every tile, a white quarter circle of radius $\tfrac12$ foot is centered at each of the four corners; the rest of the tile is shaded. Find the total shaded area, in square feet.
Givens: The floor is a rectangle $8$ ft by $10$ ft; It is fully covered by identical square tiles, each $1$ ft by $1$ ft; Each tile has four white quarter circles of radius $\tfrac12$ ft, one centered at each corner; The part of each tile not covered by white is shaded; Answer choices: (A) $80-20\pi$, (B) $60-10\pi$, (C) $80-10\pi$, (D) $60+10\pi$, (E) $80+10\pi$
Unknowns: The total shaded area of the whole floor, in square feet
Understand
Restated: A bathroom floor measures $8$ feet by $10$ feet and is covered by $1$-foot-by-$1$-foot square tiles. On every tile, a white quarter circle of radius $\tfrac12$ foot is centered at each of the four corners; the rest of the tile is shaded. Find the total shaded area, in square feet.
Givens: The floor is a rectangle $8$ ft by $10$ ft; It is fully covered by identical square tiles, each $1$ ft by $1$ ft; Each tile has four white quarter circles of radius $\tfrac12$ ft, one centered at each corner; The part of each tile not covered by white is shaded; Answer choices: (A) $80-20\pi$, (B) $60-10\pi$, (C) $80-10\pi$, (D) $60+10\pi$, (E) $80+10\pi$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #17 Visualize Spatial Relationships, #16 Change Focus / Count the Complement
The whole floor is too big to attack at once, so Tool #7 (Identify Subproblems) splits it into two easy questions: how many tiles are there, and how much of one tile is shaded? Tool #17 (Visualize Spatial Relationships) is the key unlock on a single tile — the four corner quarter circles can be imagined sliding together into one whole circle, turning an awkward four-piece white region into a single circle whose area is easy. Tool #16 (Count the Complement) then gets the shaded part the short way: instead of measuring the strange shaded shape directly, take the tile area and subtract the white circle. Multiply the per-tile shaded amount by the tile count to finish.
Execute — Answer: A
3.MD.C.7 Step 1 Count the tiles
- The floor is $8$ ft by $10$ ft, so its area is $8\times10=80$ square feet.
- Each tile is $1$ ft by $1$ ft, which is exactly $1$ square foot, so the floor holds $80$ tiles.
- Because all tiles look the same, the total shaded area will be $80$ times the shaded area of one tile.
💡 A grid of $1\times1$ squares filling an $8\times10$ rectangle simply has $8\times10$ squares.
4.NF.B.4 Step 2 Slide the quarters into one circle
- On one tile there are four white quarter circles, each of radius $\tfrac12$, one at each corner.
- Four quarter circles make one whole circle: $4\times\tfrac14 = 1$.
- Imagine sliding the four corner pieces together — they assemble into a single full circle of radius $\tfrac12$.
- So the total white on a tile is exactly one circle of radius $\tfrac12$.
💡 Four quarters of anything join back into one whole.
7.G.B.4 Step 3 Find the white area on a tile
- The area of a circle is $\pi r^2$.
- Here the radius is $r=\tfrac12$, so the white circle has area $\pi\left(\tfrac12\right)^2 = \pi\cdot\tfrac14 = \dfrac{\pi}{4}$ square feet.
- That is the white area on a single tile.
💡 Plug the radius into $\pi r^2$; halving the radius quarters the area.
7.G.B.4 Step 4 Shaded area on one tile
- The shaded region is everything on the tile that is not white.
- Rather than measure that curvy shape directly, subtract the white from the whole tile.
- The tile area is $1$ square foot and the white area is $\dfrac{\pi}{4}$, so the shaded area on one tile is $1-\dfrac{\pi}{4}$.
💡 Shaded equals the whole tile minus the white circle inside it.
6.EE.A.3 Step 5 Scale up to the whole floor
- Multiply the per-tile shaded area by the $80$ tiles.
- Distribute the $80$ across both terms: $80\times1 = 80$ and $80\times\dfrac{\pi}{4} = 20\pi$.
- So the total shaded area is $80-20\pi$ square feet, which is choice (A).
💡 Eighty identical tiles means eighty times one tile's shaded amount.
3.MD.C.7 The floor is $8$ ft by $10$ ft, so its area is $8\times10=80$ square feet. Each 4.NF.B.4 On one tile there are four white quarter circles, each of radius $\tfrac12$, one 7.G.B.4 The area of a circle is $\pi r^2$. Here the radius is $r=\tfrac12$, so the white 7.G.B.4 The shaded region is everything on the tile that is not white. Rather than measu 6.EE.A.3 Multiply the per-tile shaded area by the $80$ tiles. Distribute the $80$ across Review
Reasonableness: Check the size: $\dfrac{\pi}{4}\approx0.785$, so each tile is about $1-0.785=0.215$ square feet shaded, and $80\times0.215\approx17.2$ square feet total. The formula gives $80-20\pi\approx80-62.8=17.2$ — a match. The value is a positive number less than $80$, which makes sense because the shaded part must be smaller than the whole floor. The '$+$' choices (D) and (E) are bigger than $80$, impossible for a part of an $80$-square-foot floor, so they can be dropped immediately. Choice (C) $80-10\pi$ comes from forgetting that a full circle, not a half circle, is subtracted per tile; choice (B) $60-10\pi$ mis-sizes both the floor and the circle.
Alternative: Work the white total first, then subtract once at the end. Each tile hides one circle of area $\dfrac{\pi}{4}$, and there are $80$ tiles, so the total white area is $80\times\dfrac{\pi}{4}=20\pi$. The whole floor is $80$ square feet, so the shaded area is $80-20\pi$ — the same answer (A) with a single subtraction at the finish.
CCSS standards used (min grade 7)
3.MD.C.7Relate area to multiplication and addition operations (Finding the floor area and tile count as $8\times10=80$.)4.NF.B.4Apply and extend understanding of multiplication to multiply a fraction by a whole number (Recognizing that four quarter circles combine into one whole circle, $4\times\tfrac14=1$.)7.G.B.4Know the formulas for area and circumference of a circle (Computing the white circle area $\pi(\tfrac12)^2=\tfrac{\pi}{4}$ and subtracting it from the tile to get the shaded area.)6.EE.A.3Apply the properties of operations to generate equivalent expressions (Distributing $80$ over $1-\tfrac{\pi}{4}$ to get $80-20\pi$.)
⭐ Four corner quarter circles make one whole circle, so each tile loses $\tfrac{\pi}{4}$ to white; across all $80$ tiles the shaded floor is $80-20\pi$ square feet.
⭐ Four corner quarter circles make one whole circle, so each tile loses $\tfrac{\pi}{4}$ to white; across all $80$ tiles the shaded floor is $80-20\pi$ square feet.
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