# A volume of a tetrahedron

1. Apr 30, 2006

### MathematicalPhysicist

i have the point P(x0,y0,z0) i need to find the minimal volume of a tetrahedron which is constructed by a plane which crosses over point P, and by the axis planes.

i got that the side of the tetrahedron is sqrt[(x-x0)^2+(y-y0)^2+(z-z0)^2], but im not sure it's correct because then the answer is that the volume of the tetrahedron is 0.

2. Apr 30, 2006

### HallsofIvy

The plane that crosses the axes at (a, 0, 0), (0, b, 0), and (0, 0, c) has equation x/a+ y/b+ z/c= 1 (do you see why that's obvious?). What is the volume of that tetrahedron? Of course, to pass through the point (x0,y0,z0), it must also satisfy
x0/a+ y0/b+ z0/c= 1.

So, minimize that formula for volume of a tetrahedron (in terms of a, b, c) subject to that constraint.

3. May 1, 2006

### MathematicalPhysicist

can you tell me how did you arrive at the equation?
cause from what i can remember, you start by constructing vectors from the three points:
(a,0,-c),(a,-b,0),(0,-b,c) and then you substract them and you get the next parameter equation:
(0,-b,c)+s(-a,0,c)+t(-a,-b,2c)
and then multiply by coeffiecient vector (A,B,C), and then plug in (0,-b,c)
which i get the next equation:
x/a+y/b+z/c=0 without the 1, where did i get it wrong?