PUMaC 2015 · 几何（B 组） · 第 3 题

PUMaC 2015 — Geometry (Division B) — Problem 3

[ 4 ] For her daughter’s 12th birthday, Ingrid decides to bake a dodecagon pie in celebration. Unfortunately, the store does not sell dodecagon shaped pie pans, so Ingrid bakes a circular pie first and then trims off the sides in a way such that she gets the largest regular dodecagon possible. If the original pie was 8 inches in diameter, the area of pie that she has to trim off can be represented in square inches as aπ − b where a, b are integers. What is a + b ?

解析

[ 4 ] For her daughter’s 12th birthday, Ingrid decides to bake a dodecagon pie in celebration. Unfortunately, the store does not sell dodecagon shaped pie pans, so Ingrid bakes a circular pie first and then trims off the sides in a way such that she gets the largest regular dodecagon possible. If the original pie was 8 inches in diameter, the area of pie that she has to trim off can be represented in square inches as aπ − b where a, b are integers. What is a + b ? Solution: 64 1 It is clear that the regular dodecagon that she ended up with has radius 4. Then the area of ◦ each slice of this pie is an isosceles triangle with interior angle 30 and two legs of side length 1