Dollhouse Pizzas

Geometry Level 2

The Dollhouse Pizza Parlor sells two sizes of pizza: small-sized and doll-sized. All of their pizzas are shaped like regular octagons. If the length of the crust of each slice is 3 inches for a small slice and 1 inch for a doll slice, how many times larger is the area of a small pizza compared to a doll pizza?

3 times larger 8 times larger 9 times larger 12 times larger

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2 solutions

Akhil Bansal
Nov 6, 2015

I have compared the area of both the sectors as it is directly proportional to the area of whole octagon.
Radius of small sector and doll sector is 12 π \color{#3D99F6}{\dfrac{12}{\pi}} and 4 π \color{#D61F06}{\dfrac{4}{\pi}} respectively. [ \color{#69047E}{[} If in a circle of radius r , an arc of length l subtends an angle θ \theta radian at the centre , then we have r = l θ ] r = \dfrac{l}{\theta}\color{#69047E}{]}

Area of small sector = π 8 ( 12 π ) 2 = 18 π \dfrac{\pi}{8} \cdot \left(\color{#3D99F6}{\dfrac{12}{\pi}}\right)^2 = \dfrac{18}{\pi}
Area of doll sector = π 8 ( 4 π ) 2 = 2 π \dfrac{\pi}{8} \cdot \left(\color{#D61F06}{\dfrac{4}{\pi}}\right)^2 = \dfrac{2}{\pi}

Area of small vector Area of doll vector = 9 1 \Rightarrow \dfrac{\text{Area of small vector}}{\text{Area of doll vector}} = \dfrac{9}{1}

Zandra Vinegar Staff
Nov 6, 2015

Because the two slice sizes will be similar, 9 doll-slices can fit perfectly together, covering one small slice, therefor the whole whole area of a small pizza will be 9 times larger than the whole area of a doll pizza.

(In other words, a whole small pizza has 8 triangular slices, that could only be covered by 8 × 9 = 72 8 \times 9 = 72 doll-slices in total. 72 : 8 = 9 : 1 72:8 = 9:1 )

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