Math Labs with Activity – Volume of a Sphere Formula
To demonstrate a method to derive a formula for finding the volume of a sphere
- A hollow spherical ball of known radius
- A hollow cylinder having its height equal to twice the radius of the spherical ball and base radius equal to the radius of the spherical ball
- A few packs of table salt
- A knife
The volume of a right-circular cylinder having its base radius r and height h is given by πr²h.
Therefore, the volume of a right-circular cylinder having its base radius r and height 2r will be 2πr³.
It can be shown using the above concept that the volume of a sphere of radius r is 4/3 πr³.
Step 1: Take a hollow spherical ball of known radius r.
Cut it into two hemispheres using the knife. Take one of the hemispheres. Also, take a hollow right-circular cylinder (open at the top) having the base radius r (same as the radius of the sphere) and height equal to 2r (see Figure 41.1).
Step 2: Fill the hemisphere with salt up to the brim as shown in Figure 41.2. Pour the entire quantity of the salt from the hemisphere into the cylinder. Fill the hemisphere again with salt. Pour this salt again into the cylinder. Once again fill the hemisphere with salt and then pour this salt into the cylinder (see Figure 41.3).
Step 3: We note that this time the cylinder is completely filled with salt as shown in Figure 41.4.
Observations and Calculations
We observe that the volume of the cylinder is thrice the volume of the hemisphere.
Now, volume of the cylinder having the base radius r and height 2r = πr² x (2r)
volume of the hemisphere of radius r = 2/3 πr³.
Thus, volume of the complete sphere of radius r = 2 x (2/3 πr³)= 4/3 πr³
The volume of a sphere of radius r is given by 4/3 πr³ .