The vertices of a tetrahedron or a tetrahedral object such as methane may be inscribed in a cube as illustrated on the right. If the edge of the cube is given a value of unity, then the diagonal red line must have a value of the square root of 2 (√2) (Pythagorean Theorem). The blue line is the shortest distance (perpendicular) from the center of the cube (carbon atom?) to the face of the cube or 1/2 the length of the edge of the cube. One half of the diagonal red line is √2/2. The value of tan θ = (√2/2)/(1/2) = √2 = 1.414. The tangent with this value is arctan 1.414 = 54.73^o. But this angle is only one half the desired angle 2θ. Thus, angle 2θ equals 109.46^oor rounding off, 109.5^o. Q.E.D.

PS: The distance from the center of the cube to its corner is √3/2 (Pythagorean Theorem) or 0.87 units.