curved surface area of a cone ,
where is the radius and is the slant
height.
A frustum is formed by removing a
small cone from a similar, larger cone,
as shown below.
a) Work out, in terms of , the curved
surface area of the frustum.
b) Work out, in terms of , the total
surface area of the frustum.

To find the curved surface area of a frustum, we can start by calculating the curved surface areas of both the larger cone and the smaller cone that is removed. The formula for the curved surface area of a cone works perfectly here. If and are the radius and slant height of the larger cone and and for the smaller cone, then the curved surface area of the frustum can be computed as:

Curved Surface Area of Frustum = .

Now, for the total surface area of the frustum, you add the curved surface area to the area of the top base and the bottom base. The area of each base is:

Top Base Area =

Bottom Base Area =

So the total surface area equals:

Total Surface Area of Frustum = .

History enthusiasts might be fascinated to learn that the study of cones and their frustums has roots that go back to ancient Greek mathematicians like Archimedes! He made invaluable contributions to geometry, using conic sections to explore volumes and surface areas of solids—a practice that even inspired later developments in calculus.

For practical application, knowing how to calculate the surface area of a frustum can come in handy in various industries, especially in engineering and architecture. For instance, if you’re designing a decorative vase or a conical roof, understanding the surface area helps in estimating materials required and costs, making your beautiful creations both sturdy and financially feasible!