Example 5 The base of the pyramid is an isosceles triangle, with the base and the equal sides being 12 and 10, respectively. Moreover, the projection of the pyramid's apex onto the base lies within the triangle, and the dihedral angles between each lateral face and the base are all $30^{\circ}$. Then the height of this pyramid is $\qquad$
Analysis: From Basic Conclusion 6, we know that the projection of the apex of the cone on the base is the incenter of the base triangle. Then, given that the dihedral angle between the lateral face and the base is $30^{\circ}$, we only need to first find the inradius of the base triangle. Using plane geometry methods, it is easy to find that the inradius of the base triangle is 3, thus, the height of the pyramid is $3 \tan 30^{\circ}=\sqrt{3}$.