The slant height of a frustum of a
cone is four centimeters and the perimeters of its circular ends are 18 centimeters
and six centimeters. Find the curved surface area of the
The first thing we can do is to
remember that the frustum of a cone is the portion that remains after the top part
has been cut off by a plane that’s parallel to its base. Here, we’ve drawn a diagram of what
a frustum might look like.
Next, we’re told that the slant
height of the frustum is four centimeters. And we can mark this on our
diagram. Before doing so, we should be
careful not to confuse the slant height of our frustum marked 𝑙 with the height of
the frustum marked ℎ. We’ve been given the slant height
of the frustum which is four centimeters. Alongside this, the question has
also given us the perimeters of the circular ends. On our diagram, we’ve marked the
perimeter of the smaller circle at the top as six centimeters and the larger circle
at the base as 18 centimeters.
When given a question with circles,
it can often be a good idea to mark on the radii of these circles on your
diagrams. Here, we’ve marked the radius of
the small circle as lowercase 𝑟 and the radius of the large circle as uppercase
Now since we’ve been given the
perimeters of our circles, we can also work out the radii using the following
formula. The perimeter or circumference of a
circle is equal to two 𝜋 times its radius. For our smaller circle, we have
that six is equal to two 𝜋 times lowercase 𝑟. By dividing both sides of this
equation by two 𝜋, we get that six divided by two 𝜋 is equal to lowercase 𝑟. Six divided by two is three. And swapping this equation around,
we have that lowercase 𝑟 is equal to three over 𝜋.
We can now follow the same process
for the larger circle, which forms the base of the frustum. Doing so, we get that 18 is equal
to two 𝜋 times uppercase 𝑅. Dividing both sides again by two
𝜋, we get that 18 over two 𝜋 is uppercase 𝑅. 18 divided by two is nine. And we, therefore, have that
uppercase 𝑅 is equal to nine over 𝜋. Let’s now mark this value of the
radii of our circle onto the diagram.
Let us now look back at the
question. We’ve been asked to find the curved
surface area of the frustum. It’s important to note that we
aren’t finding the surface area of this entire 3D shape. Instead, we’ll be finding the area
of this one curved face. And we’ll be ignoring the two
circles which form the top and the base of the frustum.
Okay, now that we understand our
question, let’s outline a method that can be used. If we were to first find the curved
surface area of the large cone that made up our frustum and then subtract the curved
surface area of the section that was cut away, we would be left with the curved
surface area of our frustum. In order to use this method, we’re
going to need the slant height of this section of the cone that was cut away. And this is currently unknown to
To find this slant height, we can
use an argument based on similar triangles. If we draw a line from the center
of the base of our frustum to the tip of the cone that formed the frustum, we can
make one large right-angled triangle.
Here, we have drawn out the
triangle again to take a closer look at the information that we know. This triangle represents a slice of
the cone which has formed our frustum. The base of this triangle is the
same as the radius of the circle which forms the base of our frustum. And we’ve already calculated this
length to be nine over 𝜋.
Within our large right-angled
triangle, we can also draw another smaller right-angled triangle. This represents a slice of the
section of the cone that was cut away to form the frustum. The base of this smaller triangle
is the same as the radius of the small circle which forms the top of our
frustum. And we’ve already calculated this
to be three over 𝜋.
Now, since our large right-angle
triangle is a slice of the large cone which formed the frustum, its hypotenuse will
be the same as the slant height of the large cone. The slant height of the large cone
is formed by the slant height of the frustum which has a length of four centimeters
alongside the unknown slant height of the small cone which was cut away. And we’ll call this unknown length
𝑥. The hypotenuse of our small
right-angled triangle is, therefore, 𝑥 and of our large right-angled triangle is 𝑥
Now that we know these lengths, we
can use the fact that the small right-angled triangle has been drawn inside the
large right-angled triangle and all three angles are the same. This means that the two triangles
are similar. We know that similar triangles have
sides that scale to the same ratio.
This allows us to form an
equation. The hypotenuse of the large
right-angled triangle which is 𝑥 plus four divided by the hypotenuse of the small
right-angled triangle which is 𝑥 is equal to the base of a large right-angled
triangle, which we calculated as nine over 𝜋, divided by the base of the small
right-angled triangle, which we calculated as three over 𝜋.
Now, looking at the right-hand side
of this equation, we can see that we can cancel a factor of one over 𝜋 on the top
and the bottom half of the fraction. This leaves us with nine over three
on the right-hand side of the equation. Nine over three is of course
three. And so we can replace this. And now that we’ve done this, we
can work on solving this equation for 𝑥.
We first multiply both sides by
𝑥. And we then subtract 𝑥 from both
sides. If we then divide both sides of our
equation by two and switch things around, we find that 𝑥 is equal to four over two
which is of course two. We have now found that 𝑥 or the
hypotenuse of our small right-angled triangle has a length of two. This means that the slant height of
the small cone that was cut away from the large cone to form our frustum also has a
length of two centimeters.
Now that we know these lengths,
we’re ready to work on our curved surface areas. To do so, we can use the following
formula. The curved surface area of a cone
is equal to 𝜋 times the radius of the cone times 𝑙 which is the slant height of
the cone. Remember this is the curved surface
area for a cone.
And so in order to find the curved
surface area of our frustum, we take the curved surface area of the large cone and
subtract the curved surface area of the small cone that was cut away to form the
Here, we’ve defined that the large
cone has a radius of uppercase 𝑅 and a slant height of uppercase 𝐿 and the small
cone has a radius of lowercase 𝑟 and a slant height of lowercase 𝑙. We can now work out that the slant
height of the large cone which forms the frustum is equal to two centimeters plus
four centimeters which is of course six centimeters.
Now that we know what are our
values, let’s sub these in to the equation. The curved surface area of the
large cone is 𝜋 times nine over 𝜋 times six and the curved surface area of the
small cone is 𝜋 times three over 𝜋 times two.
Looking at both of our individual
terms, we see that we can cancel out a factor of 𝜋 and one over 𝜋 inside the
brackets. This leaves us with nine times six
take away three times two which is 54 take away six which evaluates to 48
centimeters squared. And we should remember to be
careful with our units here since we’re working with an area.
And completing this last step of
our working, we’ve answered the question. Since taking the curved surface
area of the large cone and subtracting the curved surface area of the small cone is
the curved surface area of the frustum, we have, therefore, found that the curved
surface area of the frustum is 48 centimeters squared.