Skill: Calculate the surface area of cylinders

A cylinder has three surface parts:

• Two circular bases: $2\pi r^2$
• One curved surface: $2\pi rh$

So the total surface area is

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

Here, $r$ is the radius and $h$ is the height. Surface area is measured in square units.

Example: Find the surface area of a cylinder with diameter $10\text{ cm}$ and height $12\text{ cm}$. Leave the answer in terms of $\pi$.

Step 1: Convert the diameter to a 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.

$$S=2\pi r^2+2\pi rh$$ $$S=2\pi(5)^2+2\pi(5)(12)$$

Step 3: Calculate each part.

Area of the two circular bases:

$$2\pi(5)^2=2\pi(25)=50\pi$$

Area of the curved surface:

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

Step 4: Add the areas.

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

Therefore, the surface area is

$$\boxed{170\pi\text{ cm}^2}$$

If $\pi\approx3.14$, then

$$170(3.14)=533.8$$

so the surface area is approximately $\boxed{533.8\text{ cm}^2}$. Surface area measures the outside covering of the cylinder; it is not the volume inside the cylinder.