The radius of the base of the cylinder is 6 cm, and the diagonal of the axial section is 13 cm. Find the height of the cylinder.

The axial section of the cylinder is a rectangle, one side of which coincides with the diameter of the base (d), and the other is equal to the height of the cylinder (h).
Since the radius of the base is 6 cm, then the diameter is d = 12 cm.
Consider a right-angled triangle formed by two adjacent sides of a rectangular section (d and h) of the cylinder and the diagonal of this section (D), which is the hypotenuse.
Let us write down the Pythagorean theorem for this triangle, expressing the unknown side h.
h ^ 2 = D ^ 2-d ^ 2,
h ^ 2 = 169-144,
h ^ 2 = 25,
h = 5.
Answer: 5 cm.