Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #10
Grade 7 geometry-2dPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #1 (Draw a Diagram) turns the words into a labeled picture: a big circle (the pizza) with 6 small circles lined up across its diameter. That picture immediately exposes the key length — each pepperoni's diameter — by simple division. Tool #11 (Look for Symmetry / Structure) then handles the finish: both the pizza area and the total pepperoni area are multiples of π, so the π cancels and the answer is just a clean number ratio. No algebra needed.
Find the pepperoni radius
Six pepperoni span the 12-in diameter, so each pepperoni's diameter is 12 ÷ 6 = 2 in, giving radius 1 in.
Dividing 12 in equally among 6 pepperonis is a Grade 4 'how big is each share?' division.
4.OA.A.2Draw A DiagramFind the pizza's area
The pizza's radius is 6 in, so its area is π · 6² = 36π square inches.
Plugging the radius into A = π r² is exactly the Grade 7 area-of-a-circle formula.
7.G.B.4Draw A DiagramFind the total pepperoni area
Each pepperoni has radius 1 in and area π, so 24 non-overlapping pepperoni cover 24π square inches.
Same circle-area formula, scaled by the count — the 'no overlap' rule is what lets us simply add.
The 24 non-overlapping pepperoni circles cover 24π square inches of the pizza in total.
▸ Why?
One pepperoni is a circle of radius 1 inch, and a disk of radius r covers π r² of area, so a single pepperoni covers π · 1² = π square inches.
▸ Why?
Each pepperoni is 2 inches across, and its radius is half of that, 1 inch.
▸ Why?
The six equal pepperoni widths lie edge-to-edge along the 12-inch diameter with no gaps or overlaps, so together they rebuild the whole 12 inches, and sharing 12 equally among the six makes each width 2 inches.
▸ Why?
The radius is half the width, because the diameter runs from the rim through the center to the far rim as two radii laid end to end, and both radii of one circle are the same length.
▸ Why?
A disk of radius r encloses exactly π r² of area — this is the area-of-a-circle formula.
▸ Why?
With no two pepperoni overlapping, their 24 equal areas add without double-counting, and adding π twenty-four times is 24 equal groups of π, which is 24π.
▸ Why?
Because no two pepperoni overlap, the areas they cover are separate pieces that add up with no part counted twice.
▸ Why?
Adding the same area π a total of 24 times is the same as 24 equal groups of π, and totaling equal groups is multiplication, giving 24 · π.
Take the ratio
The covered fraction is ; the π cancels top and bottom, leaving .
Recognizing that a common factor (here π) divides out of a ratio is the Grade 6 ratio-reasoning move.
6.RP.A.1Work BackwardsSimplify the fraction
Both 24 and 36 are divisible by 12, so reduces to — choice (B).
Reducing a fraction by a common factor is a Grade 4 equivalent-fractions skill.
4.NF.A.1Work BackwardsThis AMC 8 problem just needs the Grade 7 circle-area formula A = π r² — and the π cancels out, leaving a clean fraction!
- Find the pepperoni radius
- Find the pizza's area
- Find the total pepperoni area
- Take the ratio
- Simplify the fraction
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