AMC10 2006 A

AMC10 2006 A · Q24

AMC10 2006 A · Q24. It mainly tests 3D geometry (volume), Geometry misc.

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?

单位正方体的相邻面的中心连线形成一个正八面体。这个八面体的体积是多少？

(A) $\frac{1}{8}$ $\frac{1}{8}$

(B) $\frac{1}{6}$ $\frac{1}{6}$

(D) $\frac{1}{3}$ $\frac{1}{3}$

(E) $\frac{1}{2}$ $\frac{1}{2}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Two pyramids with square bases form the octahedron. The upper pyramid is shown. Since the length of $\overline{AB}$ is $\sqrt{2}/2$, the base area of the pyramid is $(\sqrt{2}/2)^2 = 1/2$. The altitude of the pyramid is $1/2$, so its volume is \[ \frac{1}{3}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{12}. \] The volume of the octahedron is $2(1/12)=1/6$.

（B）两个底面为正方形的棱锥组成八面体。图中显示的是上面的棱锥。由于 $\overline{AB}$ 的长度为 $\sqrt{2}/2$，棱锥的底面积为 $(\sqrt{2}/2)^2 = 1/2$。棱锥的高为 $1/2$，因此其体积为 \[ \frac{1}{3}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{12}. \] 八面体的体积为 $2(1/12)=1/6$。

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