Question

How do you calculate the surface area to volume ratio of a cylinder?

Answer 1

Hint: In the given question, we have been asked to find the ratio of surface area of cylinder to the volume of cylinder. In order to find the ratio, we will need to simply divide the surface area of the cylinder by the volume of the cylinder. The formula of surface area of cylinder is given by \[2\pi {{r}^{2}}+2\pi rh\] and the formula of the volume of the cylinder is given by\[\pi {{r}^{2}}h\].

Complete step-by-step solution:
We have the given 3-dimensional shape i.e. cylinder

Surface area of the cylinder is given by the formula;
Surface area of cylinder = area of the circular top + area of the circular bottom + area of the side
As we know that,
Area of circle = \[\pi {{r}^{2}}\]
Area of the side i.e. in rectangular form when we open it = \[length\times breadth\]
As we can see that the length is equal to the circumference of a circle.
Thus,
Area of the side i.e. in rectangular form when we open it = \[length\times breadth=2\pi r\times h\]
Therefore,
Surface area of cylinder = area of the circular top + area of the circular bottom + area of the side
\[\Rightarrow \pi {{r}^{2}}+\pi {{r}^{2}}+2\pi rh\]
\[\Rightarrow 2\pi {{r}^{2}}+2\pi rh\]
Surface area of the cylinder = \[2\pi {{r}^{2}}+2\pi rh\]
Now,
The volume of the cylinder is given by the formula;
Volume of a cylinder = \[\pi {{r}^{2}}h\]
So,
We have given the ratio of surface area of the cylinder to the volume of the cylinder;
\[\Rightarrow \dfrac{surface\ area\ of\ cylinder}{volume\ of\ cylinder}=\dfrac{2\pi {{r}^{2}}+2\pi rh}{\pi {{r}^{2}}h}\]
Here, ‘r’ is the radius of the cylinder and ‘h’ is the height of the cylinder.
Let take an example;
Suppose we have a cylinder of radius 6cm and height 12cm.
Then,
Surface area of the cylinder = \[2\pi {{r}^{2}}+2\pi rh=2\pi {{\left( 6 \right)}^{2}}+2\pi \times 6\times 12=72\pi +144\pi =216\pi \]
Therefore, the surface area of the cylinder is\[216\pi \].
Volume of the cylinder = \[\pi {{r}^{2}}h=\pi {{\left( 6 \right)}^{2}}\times 12=\pi \times 36\times 12=432\pi \]
Therefore, the volume of the cylinder is\[432\pi \].
Now,
The ratio of surface area of the cylinder to the volume of the cylinder;
\[\Rightarrow \dfrac{surface\ area\ of\ cylinder}{volume\ of\ cylinder}=\dfrac{2\pi {{r}^{2}}+2\pi rh}{\pi {{r}^{2}}h}=\dfrac{216\pi }{432\pi }=\dfrac{1}{2}\]

Therefore the ratio of surface area of the cylinder to the volume of the cylinder is \[\dfrac{1}{2}\].

Note: A cylinder is a three dimensional figure i.e. a figure that has length, breadth and the height. The key note is that the units for surface area would be \[c{{m}^{2}}\]or\[{{m}^{2}}\], whereas the units for the volume of the 3-D figure would be\[c{{m}^{3}}\ or\ {{m}^{3}}\]. To solve the question easily, do not put the value of \[\pi \]while calculating the ratio of surface area to volume as it will cancel out in the simplification and you will get the required ratio in an easy way as it will not complicate the answer.