### Video Transcript

In this video, we will learn how to
calculate the surface area of a sphere using either its radius or its diameter. Weโll first introduce the standard
formula for doing so and then see how we can apply this to some examples. Weโll also see how we can work backwards
from knowing the surface area of a sphere to determining its radius or diameter, which will
require some skill in solving equations.

First, we recall 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
surface area of any sphere is this, four ๐๐ squared. And we must remember that it is only the
radius that is being squared, not the factor of four or the factor of ๐. Remember that, in two dimensions, the
formula for finding the area of a circle is ๐๐ squared. So the formula for calculating the
surface area of a sphere differs only in that there is an extra factor of four.

In some cases, we may be given the
diameter rather than the radius of the sphere. Thatโs the distance between two opposite
points on the sphereโs surface, passing through the center of the sphere. In this case, we must make sure that we
calculate the radius of the sphere before attempting to find its surface area, which we can
do by recalling that the diameter is twice the radius or, equivalently, the radius is half
the diameter. Now that weโve identified the key
formulae we need, letโs have a look at some examples.

Find the surface area of the given sphere
to the nearest tenth.

Firstly, we recall the formula we need
for finding the surface area of a sphere. Itโs four ๐๐ squared, where ๐
represents the sphereโs radius. Thatโs the distance between the center of
the sphere and any point on the sphereโs surface. From the figure, we can see that the
radius of this sphere is six centimeters. So we can substitute this value of ๐
directly into our formula, giving four ๐ multiplied by six squared. And we must remember that it is only the
radius that we need to square. Now, we can simplify six squared is 36
and four multiplied by 36 is 144. So the exact surface area of this sphere
is 144๐.

However, the question asks us to give
this answer to the nearest tenth. So we can use a calculator to evaluate
this as a decimal. And it gives 452.3893. If weโre rounding to the nearest tenth,
then our deciding digit is the eight in the hundredths column which tells us that we need to
round up. So we have a value of 452.4. As the units for the radius were
centimeters, the units for the surface area of the sphere will be square centimeters. So we have our answer to the problem. The surface area of this sphere to the
nearest tenth is 452.4 square centimeters.

In this question, we saw how to find the
surface area of a sphere, given its radius. In our next example, weโll see how to
calculate the surface area of a sphere when the measurement weโve been given is its
diameter.

Find the surface area of a sphere whose
diameter is 12.6 centimeters. Use ๐ equals 22 over seven.

The formula we need for calculating the
surface area of a sphere is this, four ๐๐ squared, where ๐ is the radius of the
sphere. In this question though, we havenโt been
given the radius; weโve been given the length of the sphereโs diameter. Thatโs no great problem, though, because
we know the relationship that exists between the radius and diameter of the sphere. The radius is half the length of the
diameter. So if the diameter is 12.6 centimeters,
the radius is half of this. Thatโs 6.3 centimeters. So we can substitute this value for the
radius directly into our formula for the surface area, giving four ๐ multiplied by 6.3
squared. And remember, itโs only the radius weโre
squaring.

Now, the question actually asks us to use
22 over seven as an approximation for ๐. So this suggests we havenโt got access to
a calculator in this question. Our surface area becomes four multiplied
by 22 over seven multiplied by 6.3 squared. And letโs see how we could work this out
without a calculator. First, we can write 6.3 squared as 6.3
multiplied by 6.3. Now, we should spot that seven is a
factor of 63. So we can divide 6.3 by seven relatively
easily. Using a short division or bus-stop
method, firstly, there are no sevens in six, so we put a zero and carry the six. And then there are nine sevens in 63. So 6.3 divided by seven is 0.9. So our calculation becomes four
multiplied by 22 multiplied by 0.9 multiplied by 6.3.

We can work this calculation out in
pairs. Four multiplied by 22, first of all, is
88. And to work out 0.9 multiplied by 6.3, we
can first work out 63 times nine, which is 567, and then recall that we need to divide by 10
twice in order to give the answer to the decimal calculation. So dividing 567 by 100 gives 5.67. This also makes sense from an estimation
point of view. Weโre multiplying 6.3 by something a
little less than one, so the answer we get should be a little less than 6.3, and 5.67 is
reasonable.

Finally, we can work out 5.67 multiplied
by 88 by first working out 567 multiplied by 88 which is 49896 and then dividing this value
by 100, which gives 498.96. As the units for the diameter were
centimeters, the units for the surface area will be square centimeters. And so we have our answer to the
problem. Using 22 over seven as an approximation
for ๐, we found that the surface area of the sphere whose diameter is 12.6 centimeters is
498.96 square centimeters. Remember, the key point in this question
was that we needed to calculate the radius of the sphere first before we could calculate its
surface area.

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

What is the diameter of a sphere whose
surface area is 36๐ square centimeters?

In this question, weโve been given the
surface area of a sphere and asked to use this to determine its diameter. We recall that the general formula for
finding the surface area of a sphere is four ๐๐ squared. So by equating these two pieces of
information, we can form an equation that will enable us to determine firstly the radius of
the sphere. We have the equation four ๐๐ squared
equals 36๐. And we can now solve this equation. Firstly, we can cancel a factor of ๐ on
each side. We can then divide each side of the
equation by four to leave ๐ squared on the left-hand side and nine on the right-hand
side. So we now have the equation ๐ squared is
equal to nine.

We solve this equation by square
rooting. And weโre only going to take the positive
value here because ๐ has a physical meaning as the radius of the sphere. Nine is a square number, and its square
root is three. So we found that the radius of the sphere
is three centimeters. We must be careful, though, because it
wasnโt the radius of the sphere that we were originally asked to find. It was the diameter. But thatโs no problem because we know
that the diameter of a sphere is twice its radius. So if the radius is three, the diameter
will be six. Weโve solved the problem, and the
diameter of the sphere whose surface area is 36๐ square centimeters is six centimeters.

Letโs now take a bit of time to consider
hemispheres, which is simply half a sphere. We can therefore adapt the formula for
the surface area of a sphere to finding a formula for the surface area of a hemisphere. But we must be careful as the hemisphere
has an extra surface. In addition to the curved or lateral
surface area, which will be half the total surface area of the sphere, a hemisphere also has
an additional flat circular base, which is called the great circle of the sphere. Formally, a great circle of a sphere is
the intersection of the sphere and any plane โ โ thatโs a two-dimensional slice โ โ that
passes through the center of the sphere. But we can just think of it as any circle
which divides the sphere up into two identical hemispheres.

The curved or lateral surface area of the
hemisphere will be half the surface area of the full sphere. Thatโs half of four ๐๐ squared, which
is two ๐๐ squared. And the area of the circular base is
simply ๐๐ squared. Overall then, the total surface area of
the hemisphere is three ๐๐ squared. We must be careful in questions, though,
to make sure weโre clear whether weโve been asked to find the total surface area or simply
the curved or lateral surface area.

Find, to the nearest tenth, the curved
surface area of a hemisphere, given that the area of the great circle is 441๐ square
millimeters.

Letโs begin by sketching this hemisphere
out. Remember that the great circle of a
sphere is the circle which divides the sphere up into two hemispheres. So it is the flat circular base of our
hemisphere. Itโs this face here. Weโre told that the area of this great
circle is 441๐ square millimeters. And we also know that the general formula
for calculating the area of a circle is ๐๐ squared. So we can use these two pieces of
information to form an equation. ๐๐ squared equals 441๐. We can solve this equation to find the
radius of our hemisphere. First, we can divide through by ๐ to
give ๐ squared equals 441. We then take the square root of each side
of the equation. And as 441 is a square number, its square
root is 21. So we have that the radius of this
hemisphere is 21.

We then recall that the curved surface
area of a hemisphere is half the surface area of a full sphere. Itโs two ๐๐ squared. So we can substitute our value for the
radius into this formula. That gives two ๐ multiplied by 21
squared. We know from our earlier working that ๐
squared or 21 squared is 441. So we have two ๐ multiplied by 441 which
is 882๐. The question asks us to give this value
to the nearest tenth. So we can evaluate 882๐ on a calculator,
and it gives 2770.884. The deciding digit in this case is the
eight in the hundredths column. So weโre rounding up, which gives 2770.9,
and the units for this will be square millimeters.

Now you mayโve spotted that, actually, we
didnโt need to calculate the radius of this hemisphere at all. If the area of the great circle is ๐๐
squared and the curved surface area of the hemisphere is two ๐๐ squared, then we could
have simply doubled the area we were given for the great circle. That would give two multiplied by 441๐,
which is 882๐. And so weโd have arrived at the same
stage of calculation as we had here in our previous method. Both of these methods would be equally
fine and give the same answer of 2770.9 square millimeters.

Letโs now consider one final example with
a slightly greater emphasis on problem solving.

A water feature can be modeled as a
hemisphere with its base set onto a square patio. If the diameter of the hemisphere is four
feet and the patio has a side of length six feet, what would the visible area of the patio
be? Give your answer accurate to two decimal
places.

Letโs begin with a sketch of this water
feature. It is a hemisphere, so thatโs half a
sphere, with a diameter of four feet. And itโs sitting on a square patio, which
has a side length of six feet. The visible area of the patio will be all
of the patioโs area which isnโt covered by the circular base of this hemisphere. Itโs the area of the square minus the
area of the circle, which is in fact the great circle of this hemisphere. We know how to find the areas of each of
these two-dimensional shapes. The area of a square is simply a side
length squared; thatโs six squared. And the area of a circle is ๐๐
squared.

Now, we need to be a little careful here
because we were given the diameter of the hemisphere, which is the diameter of this
circle. Itโs four feet. So the radius is half of this; it is two
feet. We have then six squared minus ๐
multiplied by two squared. That simplifies to 36 minus four ๐. And we could leave our answer in this
form if we wanted. But this question actually asked us to
give our answer accurate to two decimal places. Evaluating on a calculator gives
23.43362, and then rounding to the required two decimal places gives 23.43. As the units for the length in this
question were given in feet, the units for area will be square feet. And so we have our answer to the
problem. The visible area of the patio to two
decimal places is 23.43 square feet.

Letโs now summarize some of the key
points that weโve seen in this video. Firstly, the surface area of a sphere can
be found using the formula four ๐๐ squared, where ๐ is the radius of the sphere. We must make sure we check carefully in
any problem whether weโve been given the radius or the diameter of a sphere. We could use the equivalent formula four
๐ multiplied by ๐ over two squared, or we could simply halve the diameter to find the
radius before substituting into the formula.

We also saw that we can work backwards
from knowing the surface area of a sphere to calculate its radius or diameter by forming and
solving an equation. We then saw that the great circle of a
sphere, which formally is the intersection of the sphere with any plane that passes through
its center, divides the sphere up into two hemispheres. A sphere has infinitely many great
circles, depending on the angle of the plane we draw. And finally, for a hemisphere, we saw
that its lateral or curved surface area is given by two ๐๐ squared and its total surface
area, which includes its circular base, is given by three ๐๐ squared.