The volume of a tetrahedron is baseArea * height / 3. If we only extend one of the edges that don’t form the base, as shown in the figure, the maximum height is obtained when the rotating face is perpendicular to the base.
Then, the height coincides with that of an equilateral triangle of side 1: sqrt(3)/2.
The area of the base is sqrt(3)/4. Therefore, the maximum volume of the tetrahedron is:
Vmax = sqrt(3)/4 * sqrt(3)/2 / 3 = 1/8 = 0.1250.
Just for fun, we can compare this volume with that of a regular tetrahedron. Its height can be calculated as follows:
h = sqrt( 12 - (2/3)2) = sqrt(5)/3
Therefore, the volume of the regular tetrahedron of unitary edge is
sqrt(3)/4 * sqrt(5)/3 / 3 = 1/12 * sqrt(5/3) = 0.1076...
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