求圓影部分的面積(10分)

2006-10-18 5:49 am

如圖，正方形的邊長為a，分別以正方形的四个頂點為圓心,邊長為半徑,在正方形内畫弧，恬好是四分之一圓,那麼這四條弧圍成的陰影部分的面積是多少?

圖
http://img180.imageshack.us/img180/8122/choh9.jpg

求方法,謝謝!!!

回答 (1)

Karin

2006-10-19 12:06 am

✔ 最佳答案

Let X^2+Y^2 = a^2 be equation of one circle,
It intersect another circle at X = a/2
When X= a/2, Y = sqrt(a^2-a^2/4)= a[sqrt(3)/2]
Because of symmetry
Another intersection point occur when X = a[sqrt(3)/2] and Y = a/2

Then angle for the segment extend from the origin is 30deg
Therefore area of shaped region
= 4 { 1/2a^2(2pi30/360) - 1/2a^2sin30 + 1/2(a[sqrt(3)/2]-a/2)^2 }
= 4 {a^2pi/12 -a^2/4 + 1/2a^2(3/4+1/4-sqrt(3)/2)}
= 4a^2 { pi/12 -1/4 + 1/2 - sqrt(3)/4 }
= a^2 {pi/3 + 1 -sqrt(3) }
= 0.315a^2

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