An equilateral cone is inscribed in the ball. What percentage of the surface of the ball is the total surface of the cone?

univerkov | May 2, 2021 | education

We denote the radius of the ball by R, the radius of the base of the cone by r, and the cone by L.

Triangle ABC, by condition, is equilateral, then L = 2 * r.

The area of the cone is: Skin = n * r * (r + L) = n * r * (r + 2 * r) = n * 3 * r ^ 2.

The area of the ball is equal to: Sball = 4 * n * R ^ 2.

To express the radius of the ball through the radius of the base of the cone, we define the area of an equilateral triangle ABC.

Sас = АС ^ 2 * √3 / 4 = L ^ 2 * √3 / 4.

The radius of the ball is the radius of the circle circumscribed around the triangle, then: R = a * b * c / 4 * Sас = L ^ 3/4 * (L ^ 2 * √3 / 4) = L / √3 = 2 * r / √3 cm.

Then Sball = 4 * n * (2 * r / √3) ^ 2 = 16 * n * r ^ 2/3.

Scon / Sball = (n * 3 * r ^ 2) / (16 * n * r ^ 2/3) = 9/16.

9 * Sball = 16 * Scon.

Sball = 16 * Scon / 9.

Let the area of the ball be 100%, then the area of the cone is 100 * 9/16 = 56.25% of the area of the ball.

Answer: The area of the cone is 56.25% of the area of the ball.

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