Question

What is the relation between surface area of sphere and lateral surface area of a right circular cylinder that just encloses the sphere?

Answer 1

Hint: First we will write the formulas for the surface area and lateral surface areas of the respective solids mentioned in the question. Then we will find the relation between the dimensions of the solids. We will take the help of ratio to find the relation.

Complete step by step solution:
Given are two solids one is a sphere and the other is a right circular cylinder.
We know that,
Surface area of the sphere is given by \[ = 4\pi {r^2}\]
And the lateral surface area of a right circular cylinder is given by \[ = 2\pi rh\]
Now the cylinder just encloses the sphere.

So we can see that the diameter of the sphere is equal to the height of the cylinder such that \[h = r + r\]

So we can write the ratio of the areas as,
\[\dfrac{{S.A{._{sphere}}}}{{L.A{._{cylinder}}}} = \dfrac{{4\pi {r^2}}}{{2\pi rh}}\]
Replace the value of h with the radius form,
\[ = \dfrac{{4\pi {r^2}}}{{2\pi r\left( {2r} \right)}}\]
On multiplying we get,
\[ = \dfrac{{4\pi {r^2}}}{{4\pi {r^2}}}\]
Thus we can say that the surface area of a sphere that is enclosed in the cylinder is equal to the lateral surface area of the cylinder.

Note:
Note that lateral surface area and total surface area are different concepts. Also a curved surface is one more concept. For a two dimensional figure we have area only whereas above two areas are for three dimensional shapes or solids we can say.