# Solving for a tetrahedron...

• Oct 17th 2012, 08:43 PM
bhill
Solving for a tetrahedron...
Hi,

I have a tetrahedron I am trying to solve. I know the angles that the three edges of the solid (that are connected to the apex) make to each other - two edges make a 45 degree angle to each other, and the third edge makes a 90 degree angle to those other two edges. I also know the length of one of the three edges - the length of the edge that makes a 90 to both of the other two edges is known.

With these three angles, and the one length - is it possible to solve for the lengths of the other two edges? And thus for the angles and length of the base?

If so, how would I proceed?

Thanks!
• Oct 17th 2012, 10:14 PM
Soroban
Re: Solving for a tetrahedron...
Hello, bhill!

Quote:

I have a tetrahedron I am trying to solve. .I know the angles that the three edges of the solid
(that are connected to the apex) make to each other. .Two edges make a 45o angle to each
other, and the third edge makes a 90o degree angle to those other two edges. .I also know
the length of one of the three edges: the edge that makes 90i to both of the other two edges.

With these three angles, and the one length, is it possible to solve for the lengths of the other
two edges? .And thus for the angles and length of the base? . No

Code:

```                      A                       o                     *  * *                   * 90o *45o*                 *        *    *             P o          *      o D             *              *    *           *                *  *         *                    * *     B o  *  *  *  *  *  *  *  o C```
We have tetrahedron $ABCD.$
$\angle BAD = 90^o,\;\angle BAC = 90^o,\;\angle CAD = 45^o$

Imagine this tetrahedron being several miles high.

Point $P$ is on edge $AB.$
And we know the distance $AP.$

Imagine cutting the tetrahedron through point $P$ with a plane.

No matter what angle we use for the cut
. . the result is a tetrahedron of the form you described.

There is not sufficient information to determine your tetrahedron.
• Oct 18th 2012, 06:40 AM
bhill
Re: Solving for a tetrahedron...
OK thanks so much Soroban.

Taking this further...If I know either one of these two angles - either angle CBA, or DBA, would I then be able to solve for the lengths of AC and AD? Or would I need both angles CBA and DBA to solve?
• Nov 30th 2012, 06:11 AM
bhill
Re: Solving for a tetrahedron...
Just wondered if anyone could help me out on this.

More specifically, what would be the minimum number of edge lengths and angle combinations required to solve this? Maybe I should give you something more specific on one of the problems I am trying to solve - different from the first...

One of the three sides of the tetrahedron is an isosceles triangle. The three edge lengths of this ISO triangle are known. The three angles of this iso triangle are also known. One of the co equal edges of the ISO triangle side shares an edge with the base. The angle that this Isosceles triangle makes to the base is also known. So I have 7 variables (I think) that are known.

Is this enough info to solve the tetrahedron?

If so how would I proceed?

If not, what is the minimum additional variables I would I need to solve it?