# Volume of tetrahedron when you are given four planes

1. Nov 28, 2007

### borovecm

1. The problem statement, all variables and given/known data
I have to find volume of tetrahedron that is bounded between 4 planes.
Planes are
x+y+z-1=0
x-y-1=0
x-z-1=0
z-2=0

2. Relevant equations
$$\vec{a}$$=$$\vec{AB}$$=(X2-X1)$$\vec{i}$$+(y2-y1)$$\vec{j}$$+(z2-z1)$$\vec{k}$$
$$\vec{b}$$=$$\vec{AC}$$=(X2-X1)$$\vec{i}$$+(y2-y1)$$\vec{j}$$+(z2-z1)$$\vec{k}$$
$$\vec{c}$$=$$\vec{AD}$$=(X2-X1)$$\vec{i}$$+(y2-y1)$$\vec{j}$$+(z2-z1)$$\vec{k}$$
V(parallelepiped)=$$\vec{a}$$$$\ast$$($$\vec{b}$$$$\times$$$$\vec{c}$$)
V(tetrahedron)=1/6*V(parallelepiped)
3. The attempt at a solution

I found four points where planes meet. These are:
A(1,0,0)
B(0,-1,2)
C(3,2,2)
D(3,-4,2)

From that I made vectors AB, AC, AD and then I put that into $$\vec{a}$$$$\ast$$($$\vec{b}$$$$\times$$$$\vec{c}$$) and got that volume of parallelepiped is 4. From there I got that volume of this tetrahedron is 2/3. Is this the correct and shortest way to get a solution? My teacher said that I can use formula V=B*v/3 but I don't know where to use it.

Last edited: Nov 29, 2007
2. Nov 28, 2007

### HallsofIvy

Staff Emeritus
I can't tell you where you would use B= B*v/2 since you haven't said what B or v mean in that formula!

3. Nov 29, 2007

### borovecm

Sorry. That's Croatian notation. I think american would be Volume=1/3*B*h where B is area of the base and h is height of tetrahedron. I can calculate h from formula for distance between point where first three planes intersect and the fourth plane. I don't know how to calculate area of the base. Is it correct that volume of this tetrahedron is 2/3?

Last edited: Nov 29, 2007
4. Nov 30, 2007

### HallsofIvy

Staff Emeritus
You can choose any 3 of the 4 vertices to be a triangular base. A quick way of finding the area is to construct vectors $\vec{u}$ and $\vec{v}$ from one of the vertices to the other two. Then the area of the base, the triangle, is $B= (1/2)|\vec{u}\times\vec{v}|$. The height of the distance from the fourth point to the plane defined by the first three points.

Last edited: Sep 22, 2011
5. Sep 22, 2011