Question

A cylinder fits inside a square prism as shown. For every cross section, the ratio of the area of the circle to the area of the square is or nr^2 or n/4. Since the area of the circle is n/4 the area of the square, the volume of the cylinder equals

Answer 1

This is the concept of ratios and proportionality;
The area of the circle is π/4, the linear scale factor will be given by:
sqrt(π/4)
=(sqrt π)/2
Therefore the volume will be given by:
(Area scale factor)^3
=[(sqrt π)/2]^3
=(π^(3/2))/8

Answer 2

Volume of cylinder =
`\pi r^2h`

Explanation:

Given : A cylinder fits inside a square prism.

To find : The volume of cylinder

Solution : Refer the attached graph.

Area of circle =
`\pi r^2`

Area of square =
`s^2`

Side of square = diameter of circle=
`D^2`

Diameter = 2r

∴ Area of square=
`2r^2=4r^2`

`(Area of circle)/(Area of Square)=(\pi r^2)/(4r^2)=(\pi)/(4)`

Area of circle is
`(\pi )/(4)` of area of square.

Volume is always = area × height

Volume of prism = Area of square × h =
`4r^2h`

Volume of cylinder = Area of circle × h =
`\pi r^2h`

Now, rate

`(Volume of cylinder)/(Volume of prism)=(\pi r^2h)/(4r^2h)=(\pi)/(4)`

⇒Volume of cylinder is
`(\pi )/(4)` of Volume of prism.

Volume of Cylinder =
`(\pi )/(4)* Volume of prism`

Volume of cylinder =
`(\pi )/(4)* 4r^2h`

Volume of cylinder =
`\pi r^2h`