Q1. Calculate the volume of a football which has a regulation diameter of 22 cm.


You’re given the diameter, so you should be able to work out straight away that the radius is 11 cm. If you don’t understand where this came from, look back at previous parts of this document. Now that we’ve established a value for the radius, it is just a matter of “plugging in” all the numbers that we have to come up with a volume.



You could also be given a value for the volume of a sphere, and be asked to establish its diameter or radius. Let’s take the previous example and “reverse engineer it” to get the radius. This requires you to know how to make the radius the subject of the formula, and this can take some practice.



If you’re not sure where this came from, it might be a good idea at this point to stop and study it and follow the steps through for yourself. Only carry on when you’re happy that you understand where the expression for the radius came from.


Plugging in our known values:


Using pi to around 9 decimal places which the computer is likely to do, gives an answer to a long set of decimal places, but I’m sure you can see that 11 is the logical conclusion.

So, what would be the surface area of a regulation football? Well this is quite simple; we know that the radius is 11 cm and so all we have to do is plug that into the equation for the surface area which is a little bit simpler than the one for volume:



Once again, and I cannot stress it strongly enough, remember your units!


Q2. Calculate the numerical value of the radius “r” of a circle and a sphere which have the same numerical value for their volume and surface area (in other words the same number, not of course the same units).


What the question is asking for is to calculate the surface area of a sphere, and the volume of another sphere where the numerical results are the same, i.e. V = SA.

The only way that this can be done is by matching the two equations together:

And then solve for “r”


If you look at the left-hand side of the equation, you should be able to see that:



The reason that I have coloured πr2 is that it is now common to both sides and can be cancelled out, this leaves us with:



All we have to do now is rearrange this expression in terms of “r”:



So the answer to our question, is that the value of “r” that provides the same numerical value for V as it does for SA is 3. Put “3” into both equations to prove it:



As you can see, the numbers are identical even though of course the units aren’t. The units cannot be identical because we are talking about 2 different entities, volume and surface area, but there is no reason why the numbers can’t be the same and in this case they are.



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