The radius of the base of the cone is 6 cm, and its generatrix is 10 cm. Find: a) the height of the cone;

The radius of the base of the cone is 6 cm, and its generatrix is 10 cm. Find: a) the height of the cone; b) axial section area; c) the total surface area of the cone

To calculate the height of the cone, consider its axial section, which has the shape of an isosceles triangle. For convenience, we will designate it as ABC. The height of the ВO splits this triangle into two rectangular ones equal to each other.

To calculate the height of the AO, we apply the Pythagorean theorem:

AB ^ 2 = BO ^ 2 + AO ^ 2;

BO ^ 2 = AB ^ 2 – AO ^ 2;

ВO ^ 2 = 10 ^ 2 – 6 ^ 2 = 100 – 36 = 64;

ВO = √64 = 8 cm.

Since the axial section is an isosceles triangle, we will find its area according to Heron’s formula:

S = √p (p – a) (p – b) (p – c).

AC = AO * 2;

AC = 6 * 2 = 12 cm;

p = (AB + BC + AC) / 2;

p = (10 + 10 + 12) / 2 = 32/2 = 16 cm;

S = √16 * (16 – 10) * (16 – 10) * (16 – 12) = √16 * 6 * 6 * 4 = √2304 = 48 cm2.

The total surface area of ​​the cone is equal to the sum of the areas of its base and lateral surface:

Sp.p. = πrl + πr ^ 2;

Sp.p. = 3.14 * 6 * 10 + 3.14 * 6 ^ 2 = 3.14 * 6 * 10 + 3.14 * 36 = 188.40 + 113.04 = 301.44 cm2.

Answer: the cone height is 8 cm, the axial cross-sectional area is 48 cm2, the total surface area is 301.44 cm2.