How do you calculate the surface area of a compound solid with a cone and a cylinder?

To calculate the surface area of a compound solid with a cone and a cylinder, find and sum their areas.

First, let's break down the compound solid into its two main parts: the cone and the cylinder. Each part has its own surface area formula. For the cylinder, the surface area consists of the curved surface area and the area of the two circular bases. The formula for the curved surface area of a cylinder is \(2\pi rh\), where \(r\) is the radius and \(h\) is the height. The area of one circular base is \(\pi r^2\), so for both bases, it is \(2\pi r^2\).

Next, for the cone, the surface area includes the curved surface area and the base. The formula for the curved surface area of a cone is \(\pi rl\), where \(r\) is the radius and \(l\) is the slant height. The base of the cone is a circle with area \(\pi r^2\).

When these two shapes are combined, the base of the cone and the top base of the cylinder overlap, so you do not count the area of the overlapping circle twice. Therefore, the total surface area of the compound solid is the sum of the curved surface area of the cylinder, the curved surface area of the cone, and the area of the bottom base of the cylinder.

In summary, the total surface area \(A\) is given by:
\[ A = 2\pi rh + \pi rl + \pi r^2 \]
This formula accounts for the curved surfaces of both the cylinder and the cone, plus the bottom base of the cylinder.