Question

The height h of a cylinder is equal to the edge of a cube. If the cylinder and the cube have the same volume, what is the radius of the cylinder?
A.\[\dfrac{h}{{\sqrt \pi }}\]
B.\[h\sqrt \pi \]
C.\[\dfrac{{\sqrt \pi }}{h}\]
D.\[\dfrac{{{h^2}}}{\pi }\]

Answer 1

Hint: First we’ll find the volume of both the figures. We have given that the volume of the cylinder and the cube are equal and the height of the cylinder and the edge of the cube is also equal using this we’ll get two equations.
Using, volume of a cylinder=$\pi {(radius)^2}height$ and Volume of cube=${\left( {edge} \right)^3}$
Therefore, using these equations we’ll have the value of the radius of the cylinder.

Complete step-by-step answer:
Given data: Height of cylinder(h)= edge of the cube
The volume of cylinder=volume of the cube
Let the edge of the cube be ‘a’ and the base radius of the cylinder be ‘r’

We, know that volume of a cylinder$ = \pi {(radius)^2}height$
Volume of cube$ = {\left( {edge} \right)^3}$
Therefore, the volume of the given cylinder$ = \pi {r^2}h$
And the volume of the cube$ = {a^3}$
According to the given data, the height of the cylinder and edge of the cube is equal
i.e. $a = h.......(i)$
and volume of both the figures are equal
i.e. volume of cylinder=volume of the cube
$ \Rightarrow \pi {r^2}h = {a^3}$
Substituting the value of ‘a’ from eq(i) , we get,
$ \Rightarrow \pi {r^2}h = {h^3}$
Dividing both sides by $\pi h$
$ \Rightarrow {r^2} = \dfrac{{{h^2}}}{\pi }$
Taking square root on both the sides
$ \Rightarrow r = \dfrac{h}{{\sqrt \pi }}$
Therefore, option(A) is correct.
Note: We don’t necessarily need the values of height and edge to solve for radius of cylinder here, we can get radius in terms of edge or length.
In this above solution we’ve used some formula regarding the cylinder and the cube.
Let us see some other formula related to the cylinder and the cube.
1.Total surface area of a cube$ = 6{(side)^2}$
2.Total surface area of the cylinder$ = 2\pi (radius)[height + radius]$
3.Curved surface area of cube$ = 4{(side)^2}$
4.Curved surface area of the cylinder$ = 2\pi (radius)(height)$