Question

1．正方体中，正四面体A₁BC₁D的体积是原正方体体积的（　　）

• A． $\frac{2}{3}$
• B． $\frac{1}{2}$
• C． $\frac{1}{3}$
• D． $\frac{1}{4}$

Answer 1

分析 正四面体的体积等于正方体的体积减去4个全等的小三棱锥的体积．

解答 解：V${\;}_{{B}_{1}-{A}_{1}B{C}_{1}}$=V${\;}_{{D}_{1}-{A}_{1}{C}_{1}D}$=V${\;}_{A-{A}_{1}BD}$=V${\;}_{C-B{C}_{1}D}$
=$\frac{1}{3}{S}_{△BCD}•C{C}_{1}$=$\frac{1}{6}$S_{正方形ABCD}•CC₁=$\frac{1}{6}$V${\;}_{ABCD-{A}_{1}{B}_{1}{C}_{1}{D}_{1}}$，
∴V${\;}_{{A}_{1}-B{C}_{1}D}$=V${\;}_{ABCD-{A}_{1}{B}_{1}{C}_{1}{D}_{1}}$-4V${\;}_{C-B{C}_{1}D}$=$\frac{1}{3}$V${\;}_{ABCD-{A}_{1}{B}_{1}{C}_{1}{D}_{1}}$．
故选：C．

点评 本题考查了棱锥的体积计算，属于基础题．