# find the top vertex coordinate of a regular tetrahedron

• February 16th 2011, 02:08 PM
jwxie
find the top vertex coordinate of a regular tetrahedron
1. The problem statement, all variables and given/known data

A regular tetrahedron has the vertices of its base A(1,1,0) B(3,1,0) C(2,1+(3^(1/2),0). Find coordinate of vertex S?

2. Relevant equations

3. The attempt at a solution
If this is a tetrahedron
http://www.kjmaclean.com/Geometry/Te...s/image004.jpg

Then we know the length by caclulating the distance formula, which gives length of 2.

My game plan is to find the height in order to determine the z-coordinate.
I thought I could just get length of the apothem, which is given by a = [sqrt(3)/6]*s, where s is 2 in this case. this gives us sqrt(3)/3

Then I tried to calculate the length of BH... so (sqrt(3)/3))^2 + (1)^2 = BH^2
and i have BH = 2/sqrt(3), or 2*sqrt(3)/3

anyway. this leads to caclulate the height AOH, and i had [2/sqrt(3)]^2 + (2)^2 = AOH^2, where 2^2 comes from the length of AB.....
and AOH is 4/sqrt(3)

I dont have any solution to this problem, but upon googling someone attempted the problem.
A regular tetrahedron has the vertices of its base A(1,1,0) B(3,1,0) C(2,1+(3^(1/2),0). Find coord of vertex S? - Yahoo! Answers

I am not sure where I did wrong, if that solution is corrected.
Just by looking at the z-value, I would have 4/sqrt(3)...... which is different from whatever was solved on yahoo answer.

• February 16th 2011, 04:03 PM
bjhopper
regular tetrahedron
Hi,
Aregular tetrahedron has four faces each face an equilateral triangle.Not long ago there was a problem answered by earboth entitled find the height of the pile in the geometry forum. If you go back you will get a nice diagram showing how to calculate the altitude given one side

bjh
• February 16th 2011, 06:46 PM
jwxie
• February 16th 2011, 07:11 PM
jwxie
this is my work
a = 2 (length of the side)
a = c

http://dl.dropbox.com/u/14655573/110216_230809.jpg

i dont have an answer to check against with
can someone plz verify this is the z-coordinate?

thanks!
• February 16th 2011, 08:12 PM
Soroban
Hello, jwxie!

Quote:

$h \,=\,\dfrac{2\sqrt{6}}{3}$

i dont have an answer to check against with
Can someone plz verify this is the z-coordinate?

Yes . . . Good work!

I get: . $D\,\!\left(2,\:1\!+\!\frac{\sqrt{3}}{3},\: \frac{2\sqrt{6}}{3}\right)$