PUMaC 2015 · 几何（A 组） · 第 1 题

PUMaC 2015 — Geometry (Division A) — Problem 1

[ 3 ] 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

[ 3 ] 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: 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 1 side length 4. The area of this slice is ( 4 )( 4 ) sin 30 = 4. So the area of the entire dodecagon 2 is 4 ⋅ 12 = 48. 2 The original area of the pie was 4 π = 16 π which means Ingrid had to cut off 16 π − 48 square inches of pie. 16 + 48 = 64 .