A cone is inscribed in the sphere, the axial section of which is an equilateral triangle with a side of 9 cm.

| May 3, 2021 | education

A cone is inscribed in the sphere, the axial section of which is an equilateral triangle with a side of 9 cm. Find the ratio of the length of the great circle of the sphere to the length of the circumference of the base of the cone.

Since the axial section of the cone is an equilateral triangle, the radius of the circle circumscribed about the triangle will be: R1 = OA = AC / √3 = 9 / √3 = 3 * √3 cm.

The diameter of the sphere is equal to the diameter of the circle, then the length of the larger circumference of the sphere is: C1 = 2 * π * ОА = 2 * π * 3 * √3 = 6 * π * √3 cm.

The radius of the circumference of the base of the cone is equal to half the length of the equilateral triangle ABC. R2 = AH = AC / 2 = 9/2 = 4.5 cm.

Then C2 = 2 * π * AH = 2 * π * 4.5 = 9 * π cm.

C1 / C2 = 6 * π * √3 / 9 * π = 2 * √3 / 3.

Answer: The ratio of the lengths of the circles is 2 * √3 / 3.



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