### Video Transcript

In this video, we will learn how to find
volumes of spheres. Weโll first introduce the standard
formula for calculating the volume of a sphere and then see how we can apply this to some
examples. Weโll also see how we can work backwards
from knowing the volume of a sphere to determining its radius or diameter, which will
require some skill in rearranging formulae.

First of all, we remember that a sphere
is a three-dimensional shape. Its size is completely determined by one
measurement, its radius, which is the distance from the center of the sphere to any point on
its surface. The formula we need for calculating the
volume of any sphere is this. Four-thirds ๐๐ cubed, where ๐ is the
radius of the sphere. We must remember that it is only the
radius that is being cubed, not the factors of four-thirds and ๐.

Although we may not require them in this
video, we can also recall some other key formulae relating to spheres and circles. Firstly, the surface area of a sphere is
four ๐๐ squared. And in two dimensions, the area of a
circle is simply ๐๐ squared. Notice that each of these formulae
includes a factor of ๐ as weโre working with circles or objects related to them. We notice also that, in the formula for
area and the formula for surface area, thereโs a factor of ๐ squared, whereas in the
formula for volume, thereโs a factor of ๐ cubed.

This makes sense when we think about the
dimensions of each of these. Area and surface area are measured in
square units. So weโre multiplying two dimensions
together, ๐ by ๐, which is ๐ squared, whereas volume is measured in cubic units. Weโre multiplying three dimensions
together, ๐ by ๐ by ๐, which is ๐ cubed.

Sometimes in a problem, we may be given
the diameter rather than the radius of the sphere. Thatโs the total distance between two
opposite points on the sphereโs surface passing through the center of the sphere. In this instance, we can calculate the
radius using the relationship between the two. The diameter is equal to twice the
radius, or equivalently the radius is equal to half the diameter.

Now that weโve identified the key
formulae weโre going to need, letโs consider some examples.

Work out the volume of the sphere, giving
your answer accurate to two decimal places.

First, we remember the formula we need
for calculating the volume of a sphere. Itโs four-thirds ๐๐ cubed, where ๐ is
the radius of the sphere. We can see from the diagram that the
radius of this sphere is 6.3 centimeters. Thatโs the distance from any point on the
sphereโs surface to the center of the sphere. So we can substitute this value of ๐
directly into our formula, giving that the volume of this sphere is equal to four-thirds ๐
multiplied by 6.3 cubed.

We must remember that it is only the
radius that we are cubing, so only the value of 6.3, not the factors of four-thirds and
๐. As weโve been asked to give our answer
accurate to two decimal places, itโs reasonable to assume that we can use a calculator. So evaluating this on our calculators
gives 1047.394424 continuing.

In order to round our answer to two
decimal places, we need to consider the value in the third decimal place. It is a four. And as this is less than five, this tells
us that weโre rounding down. So we have a value of 1047.39. The units of volume will be cubic
units. And as the units given for the radius
were centimeters, the units for the volume will be cubic centimeters. And so we have our answer to this
problem. The volume of the sphere, given to two
decimal places, is 1047.39 cubic centimeters.

Letโs now consider a problem in which we
havenโt been given the radius of the sphere but instead weโve been given its diameter.

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. Itโs four-thirds ๐๐ cubed, where ๐ is
the radius of the sphere. In this question, we havenโt been given
the radius of the sphere. Instead, weโve been told that its
diameter is 30 units. Thatโs 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 ๐
multiplied by 15 cubed. Now letโs consider how we could simplify
this answer if we didnโt 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 ๐
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 ๐ 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. Itโs one of our square numbers. 15 squared is equal to 225. So our calculation becomes 20 multiplied
by ๐ multiplied by 225. We can then think of 20 as 10 multiplied
by two. So we have 10 multiplied by two
multiplied by ๐ multiplied by 225. Multiplying two by 225 gives 450, and
then multiplying by 10 gives 4,500. So we can simplify our answer to
4,500๐.

Now notice that we didnโt use a
calculator at any point in this question. And this is called giving our answer as a
multiple of ๐. Doing so means that we can answer
questions about spheres and circles when we donโt have access to a calculator.

We werenโt given any units for the
diameter in this question. So the units for our volume will just be
general cubic units. Weโve found then that the volume of a
sphere whose diameter is 30 is 4,500๐ cubic units.

In our next example, weโll see how we can
work backwards from knowing the volume of a sphere to determining its radius or
diameter.

Find the radius of a sphere whose volume
is nine over two ๐ cubic centimeters.

In this problem, weโve been given the
volume of a sphere and asked to work backwards to determine its radius. Letโs begin by recalling the formula we
use for calculating the volume of a sphere. Itโs this, four-thirds ๐๐ cubed, where
๐ represents the radius of the sphere.

Now as weโve been given the volume and we
know the general formula for working it out, we can set these two values or expressions
equal to one another to give an equation. We have four-thirds ๐๐ cubed equals
nine over two ๐. And in order to determine the radius of
the sphere, we simply need to solve this equation.

First, we notice that thereโs a factor of
๐ on each side of the equation. So we can cancel this. Or we can think of this as dividing
through by ๐, to give four-thirds ๐ cubed equals nine over two. Next, we need to divide each side of the
equation by four-thirds in order to leave ๐ cubed on its own on the left-hand side.

We recall that dividing by a fraction is
equivalent to multiplying by the reciprocal of that fraction. So to divide by four-thirds, we can
multiply each side of the equation by three-quarters. Doing so will eliminate the factor of
four-thirds on the left-hand side, leaving just ๐ cubed. And on the right-hand side, we have nine
over two multiplied by three over four. We multiply the numerators, giving 27,
and multiply the denominators, giving eight. So we find that ๐ cubed is equal to 27
over eight. To find the value of ๐, we need to
perform the inverse or opposite of cubing, which is cube rooting. So we find that ๐ is equal to the cubed
root of 27 over eight.

Now at this point, we remember that, in
order to find the cubed root of a fraction, we can find the cubed root of the numerator over
the cubed root of the denominator. So we have that ๐ equals the cubed root
of 27 over the cubed root of eight. And these are both integer values. The cubed root of 27 is three, and the
cubed root of eight is two. So we find that the radius of this sphere
is three over two or 1.5. And as the units for the volume were
cubic centimeters, the units for the radius will be centimeters.

Now of course, we werenโt asked for it in
this problem. But if we needed to calculate the
diameter of the sphere, we just need to recall that the diameter is twice the radius. So if the radius is three over two
centimeters, then the diameter is twice this. The diameter of the sphere is three
centimeters. Weโve completed the problem there. The radius of the sphere whose volume is
nine over two ๐ cubic centimeters is three over two centimeters.

Letโs now consider how we could find the
volume of a hemisphere, which we recall is simply half of a sphere. We can therefore adapt the formula for
the volume of a sphere to give a formula for finding the volume of a hemisphere. As the volume of a sphere is given by
four-thirds ๐๐ cubed, the volume of the hemisphere with the same radius will be given by a
half multiplied by four-thirds ๐๐ cubed. Of course, we can cross-cancel a factor
of two in the denominator of the first fraction with the factor of two in the numerator of
the second, to give a simplified fraction of two-thirds. And so we find that the volume of a
hemisphere can be calculated using the formula two-thirds ๐๐ cubed.

Letโs consider one example of this.

A hemisphere has a radius of 15
inches. Work out its volume, giving your answer
in terms of ๐.

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. Itโs two-thirds ๐๐ cubed. Weโre told that the radius of this
hemisphere is 15 inches. So we can substitute this value directly
into our formula, giving two-thirds ๐ multiplied by 15 cubed.

Now weโre asked to give our answer in
terms of ๐, which suggests that we donโt 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 ๐ 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 ๐ multiplied by 15 multiplied by 15. Two times five is of course 10, and 15
multiplied by 15 we should know โ itโs one of our square numbers โ is 225. So we have 10 multiplied by ๐ multiplied
by 225. 10 multiplied by 225 is 2,250. So our value in terms of ๐ is
2,250๐.

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. Itโs 2,250๐ cubic inches.

Now we couldโve answered this problem a
slightly different way. We couldโve simply calculated the volume
of the full sphere using the formula four-thirds ๐๐ 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 2,250๐ cubic inches.

Letโs now consider one final problem in
which we introduce the definition of the great circle of a sphere.

Find, to the nearest tenth, the volume of
a sphere given that the circumference of its great circle is 90๐ inches.

So first of all, what do we mean by the
great circle of a sphere? Well, formally, it is the intersection of
the sphere and any plane โ thatโs a two-dimensional slice โ which passes through the center
of the sphere. On our diagram, this is one such great
circle. But in fact, there are infinitely many
depending on the angle of the plane we draw.

Now we know that, in order to find the
volume of any sphere, we use the formula four-thirds ๐๐ cubed. So in order to answer this question, we
need to calculate the radius of our sphere. We can do this using the information
given about the circumference of the great circle because we know that the circumference of
any circle is found using the formula ๐๐ or two ๐๐.

We can therefore form an equation using
the version of the circumference formula that involves ๐. Two ๐๐ is equal to 90๐. And we can solve this equation in order
to determine the radius of the sphere. First, we cancel a factor of ๐ from each
side, giving two ๐ equals 90. And then we divide by two to find that ๐
is equal to 45. The units for this will be inches. Finally, we can substitute this value for
the radius into our formula for the volume of a sphere, giving four-thirds multiplied by ๐
multiplied by 45 cubed.

As weโre asked to give our answer to the
nearest tenth, itโs reasonable to assume we can use a calculator to help with this
question. So evaluating on a calculator gives
121,500๐. Or as a decimal, this is equivalent to
381,703.5074. As weโre rounding to the nearest tenth,
our deciding digit is in the hundredths column. Thatโs a zero. So weโre rounding down.

So we find that the volume of the sphere
whose great circle has a circumference of 90๐ inches, to the nearest tenth, is 381,703.5
cubic inches.

Letโs now summarize some of the key
points that weโve seen in this video. Firstly, the volume of a sphere whose
radius is ๐ units is four-thirds ๐๐ cubed. And we must remember it is only the
radius that is cubed. If instead of the radius we were given
the diameter of a sphere, we need to halve it before substituting into our formula. So we must make sure we check carefully
whether itโs the radius or diameter weโve been given in each problem.

We also saw in the context of an example
that we can work backwards from knowing the volume of a sphere to calculating its radius or
diameter. And to do this, we need to form and then
solve an equation. We also saw that the volume of a
hemisphere is simply half the volume of the sphere with the same radius and can be found
using the formula two-thirds ๐๐ cubed. We also saw that the great circle of a
sphere is the intersection of the sphere with any plane that passes through the sphereโs
center. And in fact, it divides the sphere up
into two hemispheres.

Finally, we saw through our examples that
we can give our answers as appropriately rounded decimals if we have access to a
calculator. Or if we donโt have access to a
calculator or simply if an exact answer is required, then we can give our answers as
multiples of ๐.