The height of the cone is 12 cm, and the angle at the apex of the axial section is 120 °. Find the total surface area of the cone.

| April 10, 2021 | education

The axial section ABC of the cone is an isosceles triangle with an apex angle C equal to 120. The height of the OС of the cone is the height, bisector and median of the triangle ABC, then the angle ACO = AСB / 2 = 120/2 = 60.

In a right-angled triangle AOС, through the angle and leg, we determine the length of the hypotenuse and the second leg.

Cos60 = OC / AC.

AC = OC / Cos60 = 12 / (1/2) = 24 cm

tg60 = AO / OC.

AO = OC * tg60 = 12 * √3 cm.

Determine the area of ​​the base of the cone.

Sb = n * R2 = n * 432 cm2.

Let us determine the area of ​​the lateral surface of the cone.

Side = n * R * L = n * AO * AC = n * 12 * √3 * 24 = n * 288 * √3 cm2.

Then Sпов = Sсн + Sbok = n * 432 + n * 288 * √3 = 144 * (3 + 2 * √3) cm2.

Answer: The surface area of ​​the cone is 144 * (3 + 2 * √3) cm2.



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