The radius of the base of the cone is 9 dm, and the area of its axial section is 117 dm ². Find the volume of the cone.

The axial cross-sectional area of the cone is equal to half the product of the diameter of its base and the height: Ssection = 0.5 * 2r * H = r * H. From here we can find the height H:
H = Ssection / r = 117/9 = 13 dm.
The volume of the cone is equal to a third of the product of the base area Sbase by the height H: V = (1/3) * Sbase * H = (1/3) * pi * r ^ 2 * H = (1/3) * pi * 9 ^ 2 * 13 = (1/3) * pi * 81 * 13 = 351 * pi, which is approximately equal to 1102.698 dm2.