### Video Transcript

What is the volume of a sphere whose
diameter is 30?

We begin by recalling the key formula we
need for calculating the volume of a sphere. It’s four-thirds 𝜋𝑟 cubed, where 𝑟 is
the radius of the sphere. In this question, we haven’t been given
the radius of the sphere. Instead, we’ve been told that its
diameter is 30 units. That’s no great problem though because we
know the relationship between the radius and diameter of a sphere. The diameter is twice the radius, or
equivalently we can say that the radius is half the diameter.

So to find the radius of this sphere, we
simply need to halve the measurement we were given for the diameter. The radius is 30 over two, which is 15
units.

Now that we know the radius, we can
substitute directly into our formula for the volume of the sphere, giving four-thirds 𝜋
multiplied by 15 cubed. Now let’s consider how we could simplify
this answer if we didn’t have access to a calculator.

We can begin by writing 15 cubed as 15
times 15 times 15. And then we know that three is a factor
of 15. So we can cancel the three in the
denominator with one factor of three in one of the 15s in the numerator, giving four 𝜋
multiplied by five multiplied by 15 multiplied by 15. We know that we can multiply in any
order. So reordering the value slightly, this is
equivalent to four multiplied by five multiplied by 𝜋 multiplied by 15 multiplied by
15.

Now four multiplied by five we know is
equal to 20. And 15 multiplied by 15 we should be able
to work out or we should know. It’s one of our square numbers. 15 squared is equal to 225. So our calculation becomes 20 multiplied
by 𝜋 multiplied by 225. We can then think of 20 as 10 multiplied
by two. So we have 10 multiplied by two
multiplied by 𝜋 multiplied by 225. Multiplying two by 225 gives 450, and
then multiplying by 10 gives 4,500. So we can simplify our answer to
4,500𝜋.

Now notice that we didn’t use a
calculator at any point in this question. And this is called giving our answer as a
multiple of 𝜋. Doing so means that we can answer
questions about spheres and circles when we don’t have access to a calculator.

We weren’t given any units for the
diameter in this question. So the units for our volume will just be
general cubic units. We’ve found then that the volume of a
sphere whose diameter is 30 is 4,500𝜋 cubic units.