imagine you can stand all 3 sides of the tetrahedron perpendicular to the base,
as if the sides were only touching but can be pulled away from each other,
so that all 3 sides are now standing at right-angles to the base.
The vertical height "h" of any of these triangular faces in terms of the base edge length "x" is found using
Now tilt the sides until they form the tetrahedron again,
and referring to the attachment,
if we have a second blue line going to the third apex of the base not shown in the attachment (think of the 3-D picture),
then we have the triangle that forms the angle between the faces.
It's dimensions are
Then the cosine rule finds the angle between the faces
A 3-D picture would be more helpful here,
hence an oblique 3-D view is also attached,
where we need the angle between the blue lines identified by the arrows.
The angle at the base and at the apex is 60 degrees,
but the blue lines have a smaller length to the edges,
hence the angle between the faces is greater than 60 degrees.
Sorry there is a typo in the diagram.
The length of the blue line is