Find the volume of a cone with a base radius of 6cm and a generatrix of 13cm.

Consider our cone, let O be the center of the base, then consider the right-angled triangle SOA, where SO is the height of the cone, SA is the generator, OA is the radius of the base, then we find what the height of the cone will be:

SO ^ 2 = SA ^ 2 – OA ^ 2;

SO ^ 2 = 169 – 36;

SO ^ 2 = 133;

SO = √133 cm.

Let’s find the area of the base of our cone:

S = π * r ^ 2;

S = π * 6 ^ 2 = 36π cm2.

Then we can find the volume of our cone:

V = (1/3) * S * SO;

V = (1/3) * 36π * √133 = 12√133π cm3.

Answer: the volume of this cone is 12√133π cm3.