Question:

A figure formed by 3 arcs of radius 2 cm is drawn by a pair of compasses. Two identical figures of this shape are drawn on a card and then cut out. One of them is fixed and the other one rolls along the sides of the fixed figure without sliding.

Find the area (in cm2) of the region that is covered by moving figure. (Take pi = 3.14)

### 2 個解答

• 最佳解答

Let ABC be the fixed card and DEF be the moving card.

To begin with, B and E touches each other and ABED formed a straight line with length (2 + 2) = 4 cm. When EF moves along BC, D also moves until ACFD formed a straight line also with length 4 cm. Therefore, area of region covered by the moving figure

= Area of sector AEDFCA - Area of sector ABC + 2 x (area of sector DEF - area of triangle DEF).

Area of sector AEDFCA = (pi)(4)^2 x 60/360 = 8(pi)/3.

Area of sector ABC = (pi)(2)^2 x 60/360 = 2(pi)/3.

Area of sector DEF = area of sector ABC = 2(pi)/3.

Area of triangle DEF = (1/2)(2)(2)sin 60 = sqrt 3.

Therefore, Area = 8(pi)/3 - 2(pi)/3 + 4(pi)/3 - 2sqrt 3.

= 10(pi)/3 - 2sqrt 3 = 31.4/3 - 2 x 1.732 = 7 cm^2.

• but the right ans is 25.12cm^2

2010-07-16 15:05:10 補充：

but the right ans is 25.12cm^2