Re: Integer tetrahedron


Dr Tim wrote:
> Does anyone know a tetrahedron with the following properties:
> 1) The volume is a positive integer;
> 2) The edges are all integers;
> 3) None of the faces is a right-angled triangle.

I did it by trial and error.
Let A, B, C be the vertices touching the ground
Let D be the vertices up in the air

Length of AB = 18
Length of BC = 41
Length of CA = 41
Length of DA = 41
Length of DB = 41
Length of DC = 24

All edges are integers. No face is a right-angle triangle.

Let E be the mid point of AB
Triangle AEC is a right-angle triangle with edges
AE = 9;AC = 41;CE = 40
Area of ABC = 40 * 9 = 360

Let F be the mid point of DC
Triangle CFE is a right-angle triangle with edges
CF = 12;CE = 40;EF = 32
Area of CDE = 12 * 32 = 384

Let h be the height of the tetrahedron
Area of CDE = CE * h / 2 = 40 * h / 2 = 384
==> h = 19.2

Volume of the tetrahedron
= 360 * 19.2 / 3
= 2304

.