The height of the cone is 8, and the length of the generatrix is 10. find the diameter of the base of the cone.

For ease of calculation, consider the axial section of the cone. Let’s designate it ABC. This section has the shape of an isosceles triangle, the base of which is the diameter of the cone, and the lateral sides are generators.
HH is the height that divides it into two equal right-angled triangles.
Consider the triangle ΔАВН.
To calculate the radius AH, we apply the Pythagorean theorem:
AB ^ 2 = BH ^ 2 + AH ^ 2;
AH ^ 2 = AB ^ 2 – BH ^ 2;
AH ^ 2 = 10 ^ 2 – 8 ^ 2 = 100 – 64 = 36;
AH = √36 = 6 cm.
Since the base diameter is equal to two of its radii, then:
d = 2 r;
d = 2 6 = 12 cm.
Answer: The diameter of the base of the cone is 12 cm.