Video Transcript
A hemisphere has a radius of 15
inches. Work out its volume, giving your answer
in terms of 𝜋.
We recall firstly that a hemisphere is
simply half a sphere. So the formula for finding the volume of
a hemisphere is just half the formula for finding the volume of a sphere. It’s two-thirds 𝜋𝑟 cubed. We’re told that the radius of this
hemisphere is 15 inches. So we can substitute this value directly
into our formula, giving two-thirds 𝜋 multiplied by 15 cubed.
Now we’re asked to give our answer in
terms of 𝜋, which suggests that we don’t have access to a calculator for this problem. So we need to consider how to simplify
the calculation without using a calculator. We can first write 15 cubed as 15
multiplied by 15 multiplied by 15. And then as three is a factor of 15, we
can cancel the three in the denominator with a factor of three from one of the 15s in the
numerator, giving two over one 𝜋 multiplied by five multiplied by 15 multiplied by 15.
We can perform this multiplication in any
order. So perhaps the easiest is to think of it
as two times five multiplied by 𝜋 multiplied by 15 multiplied by 15. Two times five is of course 10, and 15
multiplied by 15 we should know — it’s one of our square numbers — is 225. So we have 10 multiplied by 𝜋 multiplied
by 225. 10 multiplied by 225 is 2,250. So our value in terms of 𝜋 is
2,250𝜋.
The units given for the radius were
inches. And so the units given for the volume
will be cubic inches. And so we have our answer to the
problem. It’s 2,250𝜋 cubic inches.
Now we could’ve answered this problem a
slightly different way. We could’ve simply calculated the volume
of the full sphere using the formula four-thirds 𝜋𝑟 cubed and then divided our answer by
two at the end to give the volume of a hemisphere. But of course, it would give the same
result of 2,250𝜋 cubic inches.
