SOLUTION: All the edges of a regular triangular pyramid are x units long. Find the volume of the pyramid in terms of x. In the end, a formula for volume should result. Show the steps taken t

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Question 1036471: All the edges of a regular triangular pyramid are x units long. Find the volume of the pyramid in terms of x. In the end, a formula for volume should result. Show the steps taken to get to this formula.
Answer by ikleyn(52921) About Me  (Show Source):
You can put this solution on YOUR website!
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All the edges of a regular triangular pyramid are x units long. Find the volume of the pyramid in terms of x.
In the end, a formula for volume should result. Show the steps taken to get to this formula.
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Solution

The base area of the given tetrahedron is the area of the equilateral triangle with                    
the side measure x.  So,  the base area is equal to  S%5Bbase%5D = 1%2F2.x.xsqrt%283%29%2F2 = %28x%5E2%2Asqrt%283%29%29%2F4.

Next,  let us find the measure of the height of the given tetrahedron  (Figure 1b).
The height  OP  of the given tetrahedron drops to the center  O  of its base,  which
is the intersection point of the base altitudes,  medians and angle bisectors.
It is well known fact of  Planimetry  that the intersection point of medians of a
triangle divides them in proportion  2:1  counting from the vertices  (see the lesson
Medians of a triangle are concurrent  in this site).



    Figure 1a.

Thus the segment  OA  in  Figure 1b  has the length of two third of the altitude
of the base triangle,  i.e.  |OA| = 2%2F3.%28x%2Asqrt%283%29%29%2F2 = %28x%2Asqrt%283%29%29%2F3.
Now,  the height of the pyramid is  h = sqrt%28abs%28AP%29%5E2+-+abs%28OP%29%5E2%29 = sqrt%28x%5E2+-+%28x%2Asqrt%283%29%2F3%29%5E2%29 = x%2Asqrt%281+-+1%2F3%29 = %28x%2Asqrt%282%29%29%2Fsqrt%283%29,       

and the volume of our tetrahedron is  V = 1%2F3.S%5Bbase%5D.h = 1%2F3.%28x%5E2%2Asqrt%283%29%29%2F4.%28x%2Asqrt%282%29%29%2Fsqrt%283%29 = %28x%5E3%2Asqrt%282%29%29%2F12.

Answer.  The volume of the regular tetrahedron with the edge length x is   %28x%5E3%2Asqrt%282%29%29%2F12.



    Figure 1b.