如图所示，一底面面积为S的直三棱锥被一平面截去一块，截面为A1B1C1，且AA1=a，BB1=b
如图，过A1点作平面A1B2C2∥平面ABC，
交BB1延长线于B2，交CC1于C2，交B1C1于点P
则 V(ABC-A1B1C1)=V(ABC-A1B2C2)-V(B1-A1B2P)+V(C1-A1C2P)
由图易知，上述几个体积分别如下
V(ABC-A1B2C2)=S△ABC*AA1=S*a
V(B1-A1B2P)=1/3*S△A1B2P*B1B2=1/3*S△A1B2P*(a-b)
V(C1-A1C2P)=1/3*S△A1C2P*C1C2=1/3*S△A1C2P*(c-a)
现在关键是要求出△A1B2P与△A1C2P的面积
∵B2，P，C2三点共线，∴△A1B2P与△A1C2P等高
∴S△A1B2P：S△A1C2P=B2P：C2P (1)
又∵直棱锥，∴BB1∥CC1，∴△B1B2P∽△C1C2P
即有 B2P/C2P=B1B2/C1C2=(a-b)/(c-a) (2)
又S△A1B2P+S△A1C2P=S△A1B2C2=S
∴S△A1C2P=S△A1B2C2-S△A1B2P=S-S△A1B2P (3)
将(2)(3)带入(1)，即有
S△A1B2P/(S-S△A1B2P)=(a-b)/(c-a)
可解得 S△A1B2P=(a-b)/(c-b)*S
S△A1C2P=(c-a)/(c-b)*S
带入上述体积，可得
V(B1-A1B2P)=1/3*S△A1B2P*(a-b)=1/3*(a-b)/(c-b)*S*(a-b)
V(C1-A1C2P)=1/3*S△A1C2P*(c-a)=1/3*(c-a)/(c-b)*S*(c-a)
∴几何体体积V(ABC-A1B1C1)
=V(ABC-A1B2C2)-V(B1-A1B2P)+V(C1-A1C2P)
=S*a-1/3*(a-b)/(c-b)*S*(a-b)+1/3*(c-a)/(c-b)*S*(c-a)
=S/3*[3a(c-b)-(a-b)²+(c-a)²]/(c-b)
=S/3*[(ac-ab)-b²+c²]/(c-b)
=S/3*[(c-b)*(a+b+c)]/(c-b)
=S/3*(a+b+c)
