阴影图形由 6 个大四分之一圆弧、6 个小四分之一圆弧和 2 条直线组成。求阴影部分的面积。(取 π=227\pi = \frac{22}{7}) 答:___

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完整解法

6

1. Identify the radii of the big and small quarter arcs

From the grid, the side of one square is 7 cm7\text{ cm}.
Radius of small quarter arc, r=7 cmr = 7\text{ cm}.
Radius of big quarter arc, R=2×7=14 cmR = 2 \times 7 = 14\text{ cm}.

2. Calculate the total area of the 6 big quarter circles

Area of 1 big quarter circle = 14×π×R2=14×227×142=154 cm2\frac{1}{4} \times \pi \times R^2 = \frac{1}{4} \times \frac{22}{7} \times 14^2 = 154\text{ cm}^2.
Area of 6 big quarter circles = 6×154=924 cm26 \times 154 = 924\text{ cm}^2.

3. Calculate the total area of the 6 small quarter circles

Area of 1 small quarter circle = 14×π×r2=14×227×72=38.5 cm2\frac{1}{4} \times \pi \times r^2 = \frac{1}{4} \times \frac{22}{7} \times 7^2 = 38.5\text{ cm}^2.
Area of 6 small quarter circles = 6×38.5=231 cm26 \times 38.5 = 231\text{ cm}^2.

4. Calculate the total area of the shaded figure

Total area = Area of 6 big quarter circles + Area of 6 small quarter circles
Total area = 924+231=1155 cm2924 + 231 = 1155\text{ cm}^2.
(Note: The straight lines do not enclose any additional area on their own, they just connect the arcs).

答案

1155 cm21155\text{ cm}^2

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