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How to Compare Slice Sizes on Two Pizzas Using Trigonometry

Some fraternity brothers want to order pizza — and you know how hungry college men can be. The big question is, which has bigger slices of pizza: a 12-inch pizza cut into six slices, or a 15-inch pizza cut into eight slices?

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The figure shows a 12-inch pizza and a 15-inch pizza, both of which are sliced. Can you tell by looking at them which slices are bigger — that is, have more area?

The 12-inch pizza is cut into six pieces. Each piece represents an angle of 60 degrees, which is

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radians, so you find the area of each sector (slice) by using the formula for the area of a sector using radians and putting the 6 in for the radius of the pizza with a 12-inch diameter. The answer is

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The 15-inch pizza is cut into eight pieces. Each piece represents an angle of 45 degrees, which is

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radians, so, letting the radius be 7.5 this time, the area of each sector is

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This result doesn’t tell you exactly how many square inches are in each slice, but you can see that a slice of this 15-inch pizza has an area of 7.03125π square inches, and a slice of the 12-inch pizza has an area of 6π square inches. The 15-inch pizza has bigger pieces, even though you cut it into more pieces than the 12-inch pizza. And, by the way, the difference is slightly over three square inches per slice.

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