AMC10 2005 A

AMC10 2005 A · Q20

AMC10 2005 A · Q20. It mainly tests Area & perimeter, Polygons.

An equiangular octagon has four sides of length 1 and four sides of length $\sqrt{2}/2$, arranged so that no two consecutive sides have the same length. What is the area of the octagon?

一个等角八边形有四条边长为1，四条边长为 $\sqrt{2}/2$，排列使得没有两条连续边等长。这个八边形的面积是多少？

(A) $\frac{7}{2}$ $\frac{7}{2}$

(B) $\frac{7\sqrt{2}}{2}$ $\frac{7\sqrt{2}}{2}$

(D) $\frac{4+5\sqrt{2}}{2}$ $\frac{4+5\sqrt{2}}{2}$

(E) 7 7

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) The octagon can be partitioned into five squares and four half squares, each with side length $\frac{\sqrt{2}}{2}$, so its area is $$\left(5+4\cdot\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)^2=\frac{7}{2}.$$

（A）这个八边形可以分割成五个正方形和四个半正方形，每个的边长为 $\frac{\sqrt{2}}{2}$，因此其面积为 $$\left(5+4\cdot\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)^2=\frac{7}{2}.$$

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