# Ratio of the volume and the surface area between two spheres

by Rabbani

**Question**

The radius of sphere A is twice that of a sphere B. Find the ratio among their surface areas and the ratio among their volumes.

**Answer**

Consider the spheres below. These two spheres have the radius of

**2r** and

**r** respectively.

**Sphere A Sphere B** **The ratio of the surface area of sphere A to sphere B**Let the surface area of sphere A be

**S**_{A} and the surface area of sphere B be

**S**_{B}. Now we know that the formula for the surface area of sphere

**S** is:

Therefore, to find

**S**_{A}, we just need to substitute

**r** with

**2r** and simplify the equation. This is shown below:

Similarly, to find

**S**_{B}, since the radius is

**r**, we have:

Hence, the ratio of

**S**_{A} to

**S**_{B} will be:

**The ratio of the volume of sphere A to sphere B**Let the volume of sphere A be

**V**_{A} and the volume of sphere B be

**V**_{B}. Now, we know that the formula for the volume of a sphere is:

To find

**V**_{A}, we just substitute

**r** with

**2r** and simplify the equation. This is shown below:

Next, to find

**V**_{B}, since the radius of sphere B is

**r**, we have:

Hence, the ratio of

**V**_{A} to

**V**_{B} is: