Nối MN kéo dài lần lượt cắt AB và AD tại E và F
Nối A'E cắt BB' tại P, nối A'F cắt DD' tại Q
\(\Rightarrow V_H=V_{A'.AEF}-\left(V_{PBEM}+V_{QDNF}\right)\)
CM song song FD nên theo Talet: \(\dfrac{CM}{DF}=\dfrac{CN}{DN}=\dfrac{1}{2}\Rightarrow DF=2CM=a\) \(\Rightarrow AF=2a\)
\(\Rightarrow\dfrac{DQ}{QD'}=\dfrac{DF}{A'D'}=1\Rightarrow DQ=\dfrac{a}{2}\)
Talet: \(\dfrac{EB}{CN}=\dfrac{BM}{CM}=1\Rightarrow BE=CN=\dfrac{a}{3}\) \(\Rightarrow AE=\dfrac{4a}{3}\)
\(\dfrac{BP}{B'P}=\dfrac{EB}{A'B'}=\dfrac{1}{3}\Rightarrow BP=\dfrac{1}{3}B'P=\dfrac{1}{4}BB'=\dfrac{a}{4}\)
\(V_{A'.AEF}=\dfrac{1}{3}.AA'.\dfrac{1}{2}AE.AF=\dfrac{4a^3}{9}\)
\(V_{PBEM}=\dfrac{1}{3}BP.\dfrac{1}{2}BE.BM=\dfrac{a^3}{144}\)
\(V_{QDNF}=\dfrac{1}{3}QD.\dfrac{1}{2}DN.DF=\dfrac{a^3}{18}\)
\(\Rightarrow V_H=\dfrac{55a^3}{144}\)
