In a regular tetrahedron, the height will be perpendicular to the base(and in it's center).
Think about the triangle formed by the height, a line drawn from where the height meets the base to one side, and then the altitude of that side of the tetrahedron. This is a right triangle.
The hypotenuse of our triangle is the altitude of one side. The length of the altitude of any side can be found by looking at the right triangle it forms as part of the side. We can find it's length using a 30-60-90 triangle or Pythagoras(a^2+b^2=c^2). In this case the hypotenuse would be 5cm, the side on the bottom would be 1/2 of the 5cm or 2.5 cm. Can you find the third side? I would call that length L.
Now we have one side of our triangle that includes the height of the tetrahedron (the hypotenuse).
The side on the bottom should be 1/2 of the altitude of a side or 1/2 L.
Correction: The side on the bottom is not 1/2L - the center is in the center of the base of the triangle, but that is not at 1/2 the altitude. Using a 30-60-90 triangle formed by connecting the base of the height to a corner of the base and to the midpoint of a side of the base (pictures would be so much easier), you know 1/2 of the side is 5/2 so you can use tan 30 *5/2=√3/3 * 5/2=5√3/6
We have 2 sides of our triangle now. Because it is a right triangle you can use Pythagoras again to get the 3rd side, which is the height we were looking. L^2 = (1/2L)^2+H^2. Or H^2=L^2-(1/2L)^2.
In this case L^2=5^2-(5/2)^2
h^2=75/4-75/16 correction: H^2=75/4-75/36=50/3, H=5√6/3
h^2=225/16 ignore this now
h=15/4 ignore - see corrections