Challenging geometry

2009-09-07 11:02 pm

A regular pentagon with each side = x has 5 circular arcs drawn inside it in a manner that with each vertex as a centre, from one of its adjacent vertices to the other as shown below:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Sep09/Crazyprob1.jpg

Find, in terms of x, the area of region S which is similar to (but NOT exactly as) a regular pentagon.

回答 (2)

用戶9112

2009-09-09 3:37 am

✔ 最佳答案

http://img529.imageshack.us/img529/5844/pentagon.png

圖片參考：http://img529.imageshack.us/img529/5844/pentagon.png

下次出娛樂性題目時,可能要小心一點,避免太容易在網上找到答案.
分享我的做法作參考:
先設x = 1，五角形每隻角的角度 = (5 – 2)*180 / 5 = 108
圖一,五角形面積 = 5 * (1/2)(1/2) tan54 = (5/4)tan54
亦即圖二標示各區域面積之和 = 5A + 5B + 5C + S…(1)
如圖三所示,粉紅三角為等邊,其面積為 1/2 sin60 = √3/4
等邊三角加黃色部分 = 扇形面積 = π*60/360 = π/6
黃色部分 = π/6 – √3/4
故黃色 + 粉紅 + 灰色部分面積
= √3/4 + 2*(π/6 – √3/4)
= π/3 – √3/4 = A + 2B + 3C + S … (2)
扇形面積 (黃色 + 粉紅 + 灰色 + 藍色) = π*108/360 = 3π/10 = 2A + 3B + 4C + S …(3)
(3) – (2) => A + B + C = 3π/10 – π/3 + √3/4 = √3/4 – π/30
代入(1), 5(√3/4 – π/30) + S = (5/4)tan54
S = (5/4)tan54 + π /6 – 5√3/4
S = 0.079013
中間部分面積為 0.079013x2和wiki答案相同

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Andrew

2009-09-08 8:11 am

The solution is given as follow:

圖片參考：http://i187.photobucket.com/albums/x22/cshung/7009090700640.png

參考: http://en.wikipedia.org/wiki/Pentagon

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