The distance from the center of the base of the cone to the generatrix is 3 cm. The angle at the apex of the axial

| May 23, 2021 | education

The distance from the center of the base of the cone to the generatrix is 3 cm. The angle at the apex of the axial section is 120. Find the area of the axial section of the cone.

The axial section of the cone is an isosceles triangle, the sides of which are equal to the generatrix, and the base is equal to the diameter of the base of the cone. The height drawn from the top of such a section to the base is also a bisector and divides the apex angle in half. The ratio of the distance from the center of the base of the cone to the generatrix to the height of the cone is equal to the sine of the angle between the height and the generatrix: sin α = l / h, where α = 120/2 = 60 °, l = 3 cm.

Hence, the height of the cone is h = sin α / l = sin 60/3 = 6 / √3 = 2√3.

On the other hand, the tangent of the angle between the height of the cone and the generatrix is ​​equal to the ratio of the base radius to the height of the cone: tan α = r / h, hence r = h * tan α = 2√3 * tan 60 = 2√3 * √3 = 6 cm .

The axial section area is equal to half the product of the cone height by the base diameter: Ssection = 0.5 * h * d = 0.5 * h * 2 * r = h * r = 2√3 * 6 = 12√3 ≈ 20.78 cm2 .



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