The apothem of a regular triangular pyramid is 13 cm and the height is 15 cm, find the volume.

Height cannot be more than apothem (height – leg, apothem – hypotenuse). Let’s solve the problem, provided that the height is 12 cm.
The volume of a regular triangular pyramid is found by the formula:
V = h * a² / 4√3.
We have the height according to the condition of the problem, we need to find a side.
Let’s designate this pyramid SABC, SO – pyramid height, SH – apothem.
In the right-angled triangle SOH we find the OH leg:
OH = √ (SH² – SO²) = √ (169 – 144) = √25 = 5 (cm).
OH is the radius of a circle inscribed in a regular triangle.
r = a / 2√3 → a = r * 2√3 = 10√3 (cm).
All the data is there, we find the volume:
V = h * a² / 4√3 = 12 * (10√3) ² / 4√3 = 300√3 (cm³).
Answer: the volume of the pyramid is 300√3 cm³.