The height of the cone is 6 cm, the angle at the apex of the axial section is 120. Find a) the cross-sectional area

| April 13, 2021 | education

The height of the cone is 6 cm, the angle at the apex of the axial section is 120. Find a) the cross-sectional area of the cone by a plane passing through two generatrices, the angle between which is 30 b) the area of the lateral surface

The axial section ABC of the cone is an isosceles triangle with an apex angle C equal to 1200. 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 = 600.

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 = 6 / (1/2) = 12 cm

tg60 = AO / OC.

AO = OC * tg60 = 6 * √3 cm.

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

Side = n * R * L = n * AO * AC = n * 6 * √3 * 12 = n * 72 * √3 cm2.

The section of the CКM is an isosceles triangle in which CК = CM = 12 cm, then Ssection = CM * SC * Sin30 / 2 = (12 * 12 * 1/2) / 2 = 36 cm2.

Answer: The lateral surface area is n * 72 * √3 cm2, the cross-sectional area is 36 cm2.



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