214 Part 3: The Shape of the World
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.