How do you find the volume and surface area of a cylinder?

Relevant MM Lessons:

Forthcoming

The Humble Soup Can

It is fairly common for teachers to expect us to memorize the volume and surface area formulas for right cylinders, because it is a shape we work with frequently.We will also look at the formulas for oblique (slanted) cylinders, though these are less commonly studied.

Right Cylinder - Volume

The volume of a right cylinder comes from a very similar place as the volume for any right prism. We take the base area and multiply by the height. Since the base shape is a circle, the volume formula is$$V = \pi r^2 h$$where $r$ is the radius of the base, and $h$ is the height of the cylinder.

Right Cylinder - Surface Area

Total surface area is the sum of the two circular cylinder base areas (top and bottom) and the lateral surface area.The top and bottom are each circles, so their areas are each $\pi r^2$.The lateral surface area is rectangularly shaped - just picture removing the label from a can of soup! This rectangular area has a height of $h$ (same as the cylinder) and a width of $2 \pi r$ (the cylinder bases' circumference). All together, lateral surface area is $2\pi r h$.Therefore we have

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

Oblique Cylinder - Volume

Oblique cylinders are ones that look like they've been hit by a truck:

The volume formula is identical for oblique cylinders as it is for "normal" cylinders:$$V = \pi r^2 h$$The only trap is that you must use the actual height $h$ of the slanted cylinder, not the measure of its side, $l$, which we often refer to as slant height.

Oblique Cylinder - Surface Area

The base areas are still circles, each with area $\pi r^2$. The lateral surface area is no longer a rectangle however, and cannot easily be algebraically expressed, so you probably will never be asked about it.The only way you can calculate the lateral surface area is if you know the perimeter of the ellipse that results from intersecting the oblique cylinder with a cross sectional plane on the same slanted axis.

If you know the perimeter of that ellipse is $P$, then the lateral surface area is $P \times l$, meaning that the total surface area including the top and bottom bases is$$S = 2\pi r^2 + Pl$$where again, $P$ is the cross sectional ellipse perimeter, not the base circumference.

Tips

The volume formula comes from the same (base area) $\times$ (height) concept as prisms
The lateral surface area is a rectangle with height $h$ and width $2\pi r$ (circumference) - just think about it as the label on a soup can
Total surface area is the sum of the lateral area and the two circular bases - depending on the problem you may want only the lateral area
Unless you are specifically told otherwise, I wouldn't memorize the knowledge for oblique cylinders - it is uncommonly covered and even if it is, teachers often provide the formulas