The generatrix of the cone is 8 cm; it is inclined to the base plane at an angle of 30 degrees.

The generatrix of the cone is 8 cm; it is inclined to the base plane at an angle of 30 degrees. Calculate the total surface area of the cone.

Let’s build an axial section ABC.

Triangle ABC is isosceles, AB = BC = 8 cm.

The height BO of the triangle ABC is the height of the cone and the bisector and the median of the triangle ABC.

In a right-angled triangle ABO, the angle OAB = 30, then the leg BO lying opposite this angle is equal to half the length of the hypotenuse AB.

BО = AB / 2 = 8/2 = 4 cm.

Then, by the Pythagorean theorem, AO ^ 2 = AB ^ 2 – BO ^ 2 = 64 – 16 = 48.

AO = 4 * √3 cm.

The AO leg is the radius of the circle at the base of the cone, then Ssc = π * R ^ 2 = π * 48 cm2.

Side = π * R * AB = π * 4 * √3 * 8 = π * 32 * √3 cm2.

Spov = S main + S side.

Spov = π * 48 + π * 32 * √3 = 16 * π (3 + 2 * √3) cm2.

Answer: The surface area of ​​the cone is 16 * π (3 + 2 * √3) cm2.