Find the area of the lateral surface of the cylinder if the diameter of its base is 18 cm and the diagonal of the axial section is 24 cm.

Consider the axial section of the cylinder. For convenience, we will designate it as AВСD. The AC diagonal divides it into two equal right-angled triangles. Take the ABC triangle, for example. The diameter of the BC is 18 cm, the diagonal of the AC is 24 cm, using the Pythagorean theorem we can find the height AB:
AC ^ 2 = AB ^ 2 + BC ^ 2;
AB ^ 2 = AC ^ 2 – BC ^ 2;
AB ^ 2 = 24 ^ 2 – 18 ^ 2 = 576 – 324 = 252;
AB = √252 = 15.87 cm.
The lateral surface area is equal to the product of Pi by the diameter and by the height of the cylinder:
S = πdh;
S = 3.14 * 18 * 15.87 = 896.97 cm2.
Answer: the area of the lateral surface of the cylinder is 896.97 cm2.