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

Answer:

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

Step-by-step 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=  $2r^2=4r^2$

$\frac{Area of circle}{Area of Square}=\frac{\pi r^2}{4r^2}=\frac{\pi}{4}$

Area of circle is $\frac{\pi }{4}$  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

$\frac{Volume of cylinder}{Volume of prism}=\frac{\pi r^2h}{4r^2h}=\frac{\pi}{4}$

⇒Volume of cylinder is  $\frac{\pi }{4}$  of Volume of prism.

Volume of Cylinder =$\frac{\pi }{4}\times Volume of prism$

Volume of cylinder = $\frac{\pi }{4}\times 4r^2h$

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

Answer 2

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