Skill: Calculate the surface area of cylinders

A closed cylinder has two circular bases and one curved surface.

• The two bases have total area $2\pi r^2$.
• The curved surface has area $2\pi rh$, because it can be unwrapped into a rectangle with width $2\pi r$ (the circumference) and height $h$.

Therefore, use

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

where $r$ is the radius and $h$ is the height. The answer is in square units, not cubic units.

Example: Find the surface area of a closed cylinder with diameter $10\text{ cm}$ and height $12\text{ cm}$. Give the exact answer and an approximation.

Step 1: Find the radius.

The radius is half the diameter:

$$r=\frac{10}{2}=5\text{ cm}$$

The height is

$$h=12\text{ cm}$$

Step 2: Substitute into the formula.

$$SA=2\pi(5)^2+2\pi(5)(12)$$

Step 3: Calculate each part.

Area of the two circular bases:

$$2\pi(5)^2=50\pi$$

Area of the curved surface:

$$2\pi(5)(12)=120\pi$$

Step 4: Add the areas.

$$SA=50\pi+120\pi=170\pi\text{ cm}^2$$

This is the exact surface area. Using $\pi\approx3.14$:

$$SA\approx170(3.14)=533.8\text{ cm}^2$$

So, the surface area is

$$\boxed{170\pi\text{ cm}^2\approx533.8\text{ cm}^2}$$