Question:medium

A right circular cylinder is inscribed in a sphere and the height of the cylinder is equal to the diameter of its bas Find the ratio of the volume of the sphere to that of the cylinder.

Updated On: Jan 13, 2026
  • 4:\(\sqrt{3} \)
  • 4\(\sqrt{2} \):3
  • 2\(\sqrt{2} \):1
  • 1:2
  • None of these
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The Correct Option is

Solution and Explanation

The correct answer is option (E):
None of these

Let's break down this problem step-by-step to understand why the provided options are incorrect.

We're given a right circular cylinder inscribed within a sphere. A key piece of information is that the height of the cylinder is equal to the diameter of its base. Let's use variables to represent these dimensions.

Let:

* 'r' be the radius of the cylinder's base.
* 'h' be the height of the cylinder.

Since the height equals the diameter of the base, we have h = 2r.

Now, consider how the cylinder relates to the sphere. The cylinder is *inside* the sphere. The diameter of the sphere is the longest possible straight line segment within the sphere. This diameter will be a diagonal across a cross-section of the cylinder that includes the height. This forms a right triangle.

Let 'R' be the radius of the sphere. Then the diameter of the sphere is 2R. The diagonal we described forms the hypotenuse of a right triangle with legs equal to the diameter of the cylinder's base (2r) and the height of the cylinder (h = 2r).

Using the Pythagorean theorem:

(2R)^2 = (2r)^2 + h^2

Since h = 2r, we can substitute:

(2R)^2 = (2r)^2 + (2r)^2

4R^2 = 4r^2 + 4r^2

4R^2 = 8r^2

R^2 = 2r^2

Now let's find the volumes:

* Volume of the sphere (Vs): Vs = (4/3) * pi * R^3
* Volume of the cylinder (Vc): Vc = pi * r^2 * h = pi * r^2 * (2r) = 2 * pi * r^3

We want the ratio Vs/Vc. We know that R^2 = 2r^2. We can express R in terms of r by taking the square root: R = r * sqrt(2).

Substitute this value of R in the formula for the volume of the sphere:

Vs = (4/3) * pi * (r * sqrt(2))^3 = (4/3) * pi * r^3 * 2 * sqrt(2) = (8 * sqrt(2) / 3) * pi * r^3

Now we can compute the ratio Vs/Vc:

Vs/Vc = [(8 * sqrt(2) / 3) * pi * r^3] / [2 * pi * r^3] = (8 * sqrt(2) / 3) / 2 = (4 * sqrt(2))/ 3

The ratio of the volume of the sphere to the cylinder is therefore 4 * sqrt(2) / 3, which is not represented by any of the options.

Therefore, the answer is "None of these".
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