Surface Area of Cylinders and Cones

The surface area of a 3D solid is the total area of all its outside faces. For curved shapes like cylinders and cones, we can find the surface area by imagining them "unrolled" into flat 2D shapes (called a net).

Surface Area of a Cylinder

A cylinder consists of three parts:

1. Two identical circular bases (top and bottom).
2. A curved lateral surface that, when unrolled, forms a rectangle.

The width of this rectangle is the height of the cylinder ($h$), and the length is the circumference of the circular base ($2\pi r$).

Formula: $SA = 2\pi r^2 + 2\pi rh$

• $2\pi r^2$: Area of the two circular bases.
• $2\pi rh$: Lateral (curved) surface area.

Example: Find the surface area of a cylinder with radius $r = 4$ and height $h = 7$. $SA = 2\pi (4)^2 + 2\pi (4)(7)$ $SA = 2\pi (16) + 2\pi (28)$ $SA = 32\pi + 56\pi = 88\pi$ The total surface area is $88\pi$ square units.

Surface Area of a Cone

A cone has two parts:

1. One circular base.
2. A curved lateral surface that unrolls into a sector of a circle.

To find the area of the curved part, we use the slant height ($l$), which is the distance from the edge of the base straight up the side to the tip (apex) of the cone.

Formula: $SA = \pi r^2 + \pi rl$

• $\pi r^2$: Area of the circular base.
• $\pi rl$: Lateral (curved) surface area.

Example: Find the surface area of a cone with radius $r = 3$ and slant height $l = 5$. $SA = \pi (3)^2 + \pi (3)(5)$ $SA = 9\pi + 15\pi = 24\pi$ The total surface area is $24\pi$ square units.

Special Cases: Open-Top Containers

Sometimes, a problem asks for the material needed to build an "open-top" container. This means the shape is missing its top base.

For an open-top cylinder, you only include one base instead of two. Modified Formula: $SA = \pi r^2 + 2\pi rh$

Always read the problem carefully to determine if you need the total surface area, just the lateral area, or an open-top calculation!