Video: CBSE Class X • Pack 2 • 2017 • Question 19

CBSE Class X • Pack 2 • 2017 • Question 19

07:32

Video Transcript

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 frustum.

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 𝑅.

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 us.

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 plus four.

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 frustum.

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.

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