Section 14.3: Surface Areas of Cylinders

Property

A cylinder is a three-dimensional solid with two parallel circular bases connected by a curved surface. The cylinder has three key components for surface area calculations: two circular bases each with area $\pi r^2$, and a rectangular lateral (side) surface that wraps around the cylinder. When "unrolled," this lateral surface forms a rectangle with width equal to the circumference of the base ($2\pi r$) and height equal to the cylinder's height ($h$).

Examples

• A soup can with radius 3 cm and height 10 cm has two circular bases each with area $\pi(3^2) = 9\pi$ square cm, and a lateral surface that unrolls into a rectangle with dimensions $2\pi(3) = 6\pi$ cm by 10 cm.
• A cylindrical water tank with radius 5 feet and height 8 feet consists of two circular ends each with area $25\pi$ square feet, plus a curved side surface with circumference $10\pi$ feet wrapping around the 8-foot height.
• A paper towel tube with radius 2 inches and height 12 inches has circular ends with area $4\pi$ square inches each, and if you cut and unroll the cardboard tube, it forms a rectangle that is $4\pi$ inches wide and 12 inches tall.

Explanation

Understanding cylinder structure is essential for calculating surface area. Think of a cylinder as being made from two circular "caps" plus a rectangular piece of material wrapped around to form the sides. The key insight is that the curved lateral surface, when flattened out, becomes a rectangle whose width matches exactly the circumference of the circular base.

Property

Cylinder with radius $r$ and height $h$:
Surface Area $= 2\pi r^2 + 2\pi rh$
This consists of:

• Two circular bases: $2\pi r^2$
• Curved lateral surface: $2\pi rh$

Examples

• A cylinder with radius 3 cm and height 8 cm has a Surface Area of $2\pi(3^2) + 2\pi(3)(8) = 2\pi(9) + 2\pi(24) = 18\pi + 48\pi = 66\pi$ square cm, which is approximately $207.3$ square cm.
• A cylindrical can with radius 2 inches and height 5 inches has a Surface Area of $2\pi(2^2) + 2\pi(2)(5) = 8\pi + 20\pi = 28\pi$ square inches, which is approximately $87.9$ square inches.
• A cylinder with radius 4 ft and height 6 ft has a Surface Area of $2\pi(4^2) + 2\pi(4)(6) = 32\pi + 48\pi = 80\pi$ square ft, which is approximately $251.3$ square ft.

Explanation

The surface area of a cylinder measures the total area of all its outside surfaces. This includes the two circular bases (top and bottom) plus the curved surface that wraps around the side. Think of it as the amount of material needed to completely cover the cylinder, like wrapping paper or paint.

Property

When a cylinder is scaled by a length factor of $k$, its surface area is scaled by a factor of $k^2$.

If the original cylinder has surface area $SA$, then the scaled cylinder has surface area $k^2 \cdot SA$.