Basic Math and Pre-Algebra

PART 3. The Shape of the World

 

CHAPTER 16. Surface Area and Volume

 

Most of the geometry we learn is plane geometry, which is geometry in two dimensions. But you don’t live in a two-dimensional world. For life in three dimensions, you need to look at figures in space, not just on a flat surface or plane. Those 3-D figures are commonly referred to as solids (even if they’re hollow).

The figures referred to as solids break down into several categories. There are figures with circular bases, figures with bases that are polygons, and figures that don’t have a base. Those first two categories each divide into the objects that come to a point and those that don’t. You’ll get all those shapes organized and assign names to each. And you’ll learn how to find the two most common measurements for each of them: volume and surface area.

 

Measuring Solids

 

Line segments have length, a number that tells you the distance from one endpoint to the other. Length is measured in linear units, like inches, feet, centimeters, and meters. Plane figures like polygons and circles have two important measurements. The distance around the figure, perimeter for polygons and circumference for circles, is a measure of length, so it too is measured in linear units.

When you start measuring area, the space contained within the figure, you’re not just measuring length. You’re using measurements of length and width, working in two dimensions rather than one, so the area is measured in square units. The “square” designation makes sense if you remember you multiplied inches times inches or meters times meters to find that area. It makes sense that it’s measured in square inches or square meters.

And now you want to measure three-dimensional figures. What kind of units will you need? That depends on what you’re measuring. The length of an edge or the diameter of a circle that is a base of a 3-D figure will still be measured in linear units. If you want to talk about the surface area of a solid, that, like any area, is going to be measured in square units. When you begin to talk about volume, the measure of the amount this 3-D object might hold or the measure of the space it takes up, you’re talking about length and width and height. That demands a new unit, a 3-D unit, called a cubic unit.

 

DEFINITION

The surface area of a solid is the total of the areas of all the faces.

The volume of a solid is the measure of the space contained by the solid.

 

The simplest image of what we measure when we talk about volume is to imagine you have blocks, each 1 unit wide, 1 unit long, and 1 unit high. You can put your blocks into a box, packing them in rows, row after row until the bottom is full. Then you start making another layer of blocks on top of that. And another, and another, until the box is full.

Each of your blocks is a cube 1 unit by 1 unit by 1 unit. It’s one unit cubed, or 1 cubic unit. The number of blocks you packed into the box is the volume of the box. It’s how many blocks the box will hold. It’s the volume of the box, in cubic units.

The problem, of course, is that you can’t always find volumes by packing cubes into boxes. It takes too long, you don’t always have enough cubes, and the cubes don’t always fit nicely into the container. So you need other strategies for finding volume. Let’s look at the different kinds of solids and work out strategies for each one.