# Surface Area and Volume

### Topics

Rather than busting open 3D solids like they're set pieces in a Hollywood movie, we can use **lateral faces** and **lateral edges** to find the surface area.

A **lateral face** is any face that's not a base. But while a base is still a face, a lateral face is not a base. Is this the place for our rhyming grace? We don't think that this is the case. A **lateral edge** is the intersection of two lateral faces. Sounds either germy or painful.

The **altitude** of the prism is its height. Well why don't they just say that, then?

Sometimes, the lateral edge of a prism and the altitude are the same. That's called a **right prism** because the lateral edge makes a right angle with the base. An **oblique prism** doesn't do that, which is just not right.

The **lateral area** is the surface area of only the lateral faces. The lateral area *L* of all prisms makes a big rectangle. The lateral area will be a rectangle with an altitude of height *h* and a width equal to the perimeter of the base *P*. It doesn't matter if the base looks like this...

…Or this…

…Or even this…

### Sample Problem

What is the lateral surface area of a regular heptagonal prism with a height of 15 feet and a base edge of 15 feet?

Since the lateral area *L* equals the height *h* times the perimeter of the base *P*, we have the following.

*L* = *Ph**L* = (7 × 15 ft)(15 ft)*L* = 1575 ft^{2}

This same rule applies to cylinders, too. Cylinders are just like prisms except the base is a circle instead of a polygon.

The perimeter of a circle is actually a knight named Sir Cumference.

*L* = *PhL* = (Sir Cumference)

*h*

Sir Cumference was a fairly round fellow who really liked pie. In fact, he liked it so much that he ordered two but ate the *e*'s so now he has 2π.

A circle's circumference equals 2π times the radius. So replace the perimeter with that, and we have the formula for the lateral area of a cylinder.

*L* = 2π*rh*

### Sample Problem

The Catfish Factory sells tuna, but is run by cats (because honestly, who else would work in that stink?). They need to figure out how big to make their labels that go around the cans of tuna that are 2 inches tall and have diameters of 8 inches.

The labels will go around the entire lateral area of the can. We'll start with the lateral area formula for cylinders:

*L* = 2π*rh*

The height of the can is 2 inches and the radius, which is half the diameter, is 4 inches.

*L* = 2π(4 in)(2 in)*L* = 16π in^{2}*L* ≈ 50.3 in^{2}

Now that they know how big to make their labels, the cats can sell (or lick) their cans of tuna in peace. It doesn't really matter since no one's going to buy canned tuna made by cats anyway. Can you say hairball?