How to Compare Slice Sizes on Two Pizzas Using Trigonometry
You can use trigonometry to measure different parts of a circle. For example, say you want to order pizza, but you’re not sure which size to get. You need to know which pizza has bigger slices: a 12-inch pizza cut into six slices, or a 15-inch pizza cut into eight slices.
The following figure shows a 12-inch and a 15-inch pizza that 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 π/3 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 left with the multiplier of π just so you can compare the sizes between the two pizzas — they’ll both have a multiplier of π in them.)
The 15-inch pizza is cut into eight pieces. Each piece represents an angle of 45 degrees, which is π/4 radians, so, letting the radius be 7.5 this time, the area of each sector is
The π/3 in this formula needs to be π/4.
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.