Chapter 9Basic Geometry
9-1 Introducing Geometric Figures
Geometry is the study of shapes, namely points, lines, angles, surfaces, and solids. Why should we study geometry? There are a lot of practical applications that are mostly concerned with measuring or calculating the size of things — everything from measuring our waistlines to figuring out the number of rolls of wallpaper required for fixing up the living room. In addition to endless practical applications, geometry also helps us train our minds to think analytically about all kinds of problems, even problems that do not seem to be directly related to mathematics.
Points and Lines
Basic geometry is built upon a foundation of points, lines, and line segments
Points have no dimensions. They are purely imaginary and invisible. But we need to work with them, because we need them to define lines, and we need lines to represent figures we want to measure and analyze. So we usually indicate a point with a dot.
What is a line? Technically speaking, a line is made up of a countless number of points that are lined up in a straight row. Since points are imaginary, invisible things, it figures that a line is also an imaginary, invisible thing. We usually indicate a line with a ... well, a line.
Technically speaking once again, a line has no beginning or end. The imaginary, invisible line stretches out to infinity in both directions. Such a thing has no practical application in the real-world, so we draw lines on paper, on a computer screen, or in the sand. And we give them a starting point and and ending point that we can clearly see. A line that we give a starting point and ending point is called a line segment.
A plane is an imaginary, invisible flat surface that extends indefinitely in two dimensions. In order to make any practical use of planes, we must apply some limits. We can use some points, lines, and curves to create those limits. And there are six basic ways to combine points, lines, and curves to make sense of the notion of plane figures: triangle, square, rectangle, parallelogram, trapezoid, and circle. There are far more actual plane figures than anyone would care to count; however, they are all made up from variations and combinations of these six basic plane figures.
The six plane figures of basic geometry.
More About TrianglesA triangle is a plane figure that consists of three sides and three angles. All three sides and all three angles may be equal, all three sides and all three angles may be unequal, and there are all sorts of combinations of equal and unequal sides and angles. In other words, there are a lot of different kinds of triangles.
Here are some samples of different kinds of triangles. Although they might appear to be different in many respects they have these features in common:
More About Squares and RectanglesSquares and rectangles are plane figures that have four sides and four right angles.
More About Trapezoids and ParallelogramsA parallelogram is a very close cousin of the rectangle it is a plane figure with four sides, and the opposite sides are parallel. The big difference is that the angles of a trapezoid are not right angles.
A trapezoid is a closed, 4-sided plane figure that has just two parallel sides. This makes the figure quite different from rectangles and parallelograms that have two sets of parallel sides.
More About CirclesMost people recognize a circle when they see one. And they can draw circles, too. But few people can give a good mathematical definition of a circle. Here is a suitable mathematical definition:
A circle is the set of points that are all the same distance from a single point at the center of the circle.
The distance from the center of a circle to its outer edge is called the radius.
You have seen that plane figures are built from points and lines that are arranged in certain ways in a flat, two-dimensional world of length and width. Now you can extend that idea to solid figures. Solid figures are built from points, lines, and plane figures that are arranged in three-dimensional space--a world that has length, width, and depth. Here are the solid versions of the six basic plane figures described in the previous section of this lesson.
Six solid figures of basic geometry.