Question

What is the surface area to volume ratio of a sphere?

Answer 1

Hint: This type of question depends on the concept of finding volume and surface area of a sphere. Here, we first find the volume and surface area of the sphere and then take the ratio. We know that the volume of a sphere is given by \[\dfrac{4\pi {{r}^{3}}}{3}\] while formula for calculating surface area of a sphere is \[4\pi {{r}^{2}}\] where \[r\] represents the radius of the sphere.

Complete step by step solution:
Now we have to find out the ratio of surface area to volume of a sphere.
We know that the surface area of a sphere with radius \[r\] is given by,
\[\Rightarrow \text{Surface Area = }4\pi {{r}^{2}}\]
Also we know that the formula for the volume of a sphere is
\[\Rightarrow \text{Volume = }\dfrac{4\pi {{r}^{3}}}{3}\]
Hence, we can write ratio of surface area to volume is equal to
\[\Rightarrow \dfrac{\text{Surface Area}}{\text{Volume}}=\text{ }\left( \dfrac{4\pi {{r}^{2}}}{\left( \dfrac{4\pi {{r}^{3}}}{3} \right)} \right)\]
\[\Rightarrow \dfrac{\text{Surface Area}}{\text{Volume}}=\left( \dfrac{4\pi {{r}^{2}}}{\dfrac{4}{3}\pi {{r}^{3}}} \right)\]
We can simplify it further,
\[\Rightarrow \dfrac{\text{Surface Area}}{\text{Volume}}=4\times \left( \dfrac{3}{4} \right)\times \left( \dfrac{\pi }{\pi } \right)\times \left( \dfrac{{{r}^{2}}}{{{r}^{3}}} \right)\]
Thus we can obtain the final ratio as
\[\Rightarrow \dfrac{\text{Surface Area}}{\text{Volume}}=\dfrac{3}{r}\]
\[\Rightarrow \text{Surface Area : Volume = }3:r\]

Hence the surface area to volume ratio of a sphere is \[3:r\]

Note: In this type of problem students may make mistakes in writing formulas for surface area as well as volume of sphere. Also students have to take care in calculation of ratio as we have to find surface area to volume ratio so we must divide surface area by the volume of the sphere. In case of simplification also we have to separate out each and every term which will help us in cancellation of common terms easily.