Basic Math and Pre-Algebra For Dummies (75 page)

So this fraction simplifies as follows:

• = 3
  

When you understand how to cancel out units in fractions and how to set up fractions equal to 1 (see the preceding sections), you have a foolproof system for converting units of measurement.

Suppose you want to convert 7 meters into feet. Using the equation 1 meter = 3.26 feet, you can make a fraction out of the two values, as follows:

Both fractions equal 1 because the numerator and the denominator are equal. So you can multiply the quantity you're trying to convert (7 meters) by one of these fractions without changing it. Remember that you want the meters unit to cancel out. You already have the word
meters
in the numerator (to make this clear, place 1 in the denominator), so use the fraction that puts
1 meter
in the denominator:

Now cancel out the unit that appears in both the numerator and the denominator:

At this point, the only value in the denominator is 1, so you can ignore it. And the only unit left is
feet,
so place it at the end of the expression:

Now do the multiplication (Chapter
11
shows how to multiply decimals):

It may seem strange that the answer appears with the units already attached, but that's the beauty of this method: When you set up the right expression, the answer just appears.

You can get more practice converting units of measurement in Chapter
18
, where I show you how to set up conversion chains and tackle word problems involving measurement.

Chapter 16

In This Chapter

Knowing the basic components of geometry: points, lines, angles, and shapes

Examining two-dimensional shapes

Looking at solid geometry

Finding out how to measure a variety of shapes

G eometry is the mathematics of figures such as squares, circles, triangles, and lines. Because geometry is the math of physical space, it's one of the most useful areas of math. Geometry comes into play when measuring rooms or walls in your house, the area of a circular garden, the volume of water in a pool, or the shortest distance across a rectangular field.

Although geometry is usually a yearlong course in high school, you may be surprised by how quickly you can pick up what you need to know about basic geometry. Much of what you discover in a geometry course is how to write geometric proofs, which you don't need for algebra — or trigonometry, or even calculus.

In this chapter, I give you a quick and practical overview of geometry. First, I show you four important concepts in plane geometry: points, lines, angles, and shapes. Then I give you the basics on geometric shapes, from flat circles to solid cubes. Finally, I discuss how to measure geometric shapes by finding the area and perimeter of two-dimensional forms and the volume and surface area of some geometric solids.

Of course, if you want to know more about geometry, the ideal place to look beyond this chapter is
Geometry For Dummies
, 2nd Edition, by Mark Ryan (published by Wiley)!

Plane geometry
is the study of figures on a two-dimensional surface — that is, on a plane. You can think of the
plane
as a piece of paper with no thickness at all. Technically, a plane doesn't end at the edge of the paper — it continues forever.

In this section, I introduce you to four important concepts in plane geometry: points, lines, angles, and shapes (squares, circles, triangles, and so forth).

A
point
is a location on a plane. It has no size or shape. Although in reality a point is too small to be seen, you can represent it visually in a drawing by using a dot.

When two lines intersect, as shown in this figure, they share a single point. Additionally, each corner of a polygon is a point. (Keep reading for more on lines and polygons.)

A
line
— also called a
straight line
— is pretty much what it sounds like; it marks the shortest distance between two points, but it extends infinitely in both directions. It has length but no width, making it a one-dimensional (1-D) figure.

Given any two points, you can draw exactly one line that passes through both of them. In other words, two points
determine
a line.

When two lines intersect, they share a single point. When two lines don't intersect, they are
parallel,
which means that they remain the same distance from each other everywhere. A good visual aid for parallel lines is a set of railroad tracks. In geometry, you draw a line with arrows at both ends. Arrows on either end of a line mean that the line goes on forever (as you can see in Chapter
1
, where I discuss the number line).

A
line segment
is a piece of a line that has endpoints, as shown here.

A
ray
is a piece of a line that starts at a point and extends infinitely in one direction, kind of like a laser. It has one endpoint and one arrow.

An
angle
is formed when two rays extend from the same point.

Angles are typically used in carpentry to measure the corners of objects. They're also used in navigation to indicate a sudden change in direction. For example, when you're driving, it's common to distinguish when the angle of a turn is “sharp” or “not so sharp.”

The sharpness of an angle is usually measured in
degrees.
The most common angle is the
right angle
— the angle at the corner of a square — which is a 90° (90-degree) angle: