Getting to Know the Five Simplest Geometric Objects
The study of geometry begins with the definitions of the five simplest geometric objects — point, line, segment, ray, and angle — as well as two extra definitions (plane and 3-D space) that are thrown in for no extra charge. Collectively, these terms take you from no dimensions up to the third dimension.
Here are the definitions of segment, ray, angle, plane, and 3-D space and the “undefinitions” of point and line (these two terms are technically undefined):
Point: A point is like a dot except that it actually has no size at all; or you can say that it’s infinitely small (except that even saying infinitely small makes a point sound larger than it really is). Essentially, a point is zero-dimensional, with no height, length, or width, but you draw it as a dot, anyway. You name a point with a single uppercase letter, as with points A, D, and T in the following figure.
Line: A line is like a thin, straight wire (although really it’s infinitely thin — or better yet, it has no width at all). Lines have length, so they’re one-dimensional. Remember that a line goes on forever in both directions, which is why you use the little double-headed arrow as in
(read as line AB).
Check out the above figure again. Lines are usually named using any two points on the line, with the letters in any order. So
is the same line as
Occasionally, lines are named with a single, italicized, lowercase letter, such as lines f and g in the figure..
Line segment (or just segment): A segment is a section of a line that has two endpoints. See the above figure yet again. If a segment goes from P to R, you call it segment PR and write it as
You can also switch the order of the letters and call it
Segments can also appear within lines, as in
Note: A pair of letters without a bar over it means the length of a segment. For example, PR means the length of
Ray: A ray is a section of a line (kind of like half a line) that has one endpoint and goes on forever in the other direction. If its endpoint is point K and it goes through point S and then past it forever, you call the “half line” ray KS and write
See the above figure.
The first letter always indicates the ray’s endpoint. For instance,
can also be called
because either way, you start at A and go forever past B and C.
however, is a different ray.
Angle: Two rays with the same endpoint form an angle. Each ray is a side of the angle, and the common endpoint is the angle’s vertex. You can name an angle using its vertex alone or three points (first, a point on one ray, then the vertex, and then a point on the other ray).
Check out the above figure.
Angles can also be named with numbers, such as the angle on the right in the figure, which you can call
The number is just another way of naming the angle; it has nothing to do with the size of the angle.
The angle on the right also illustrates the interior and exterior of an angle.
Plane: A plane is like a perfectly flat sheet of paper except that it has no thickness whatsoever and it goes on forever in all directions. You might say it’s infinitely thin and has an infinite length and an infinite width. Because it has length and width but no height, it’s two-dimensional. Planes are named with a single, italicized, lowercase letter or sometimes with the name of a figure (a rectangle, for example) that lies in the plane. The above figure shows plane m, which goes out forever in four directions.
3-D (three-dimensional) space: 3-D space is everywhere — all of space in every direction. You could start with an infinitely big map that goes forever to the north, south, east, and west. That’s a two-dimensional plane. Then to get 3-D space from this map, you’d add the third dimension by going up and down forever.
There’s no good way to draw 3-D space (the above figure shows one attempt, but it’s not going to win any awards). Unlike a box, 3-D space has no shape and no borders.
Because 3-D space takes up all the space in the universe, it’s sort of the opposite of a point, which takes up no space at all. But on the other hand, 3-D space is like a point in that both are difficult to define because both are completely without features.
Here’s something a bit peculiar about the way objects are depicted in geometry diagrams: Even if lines, segments, rays, and so on, don’t appear in a diagram, they’re still sort of there — as long as you’d know where to draw them. For example, the first figure contains a segment,
that goes from P to D and has endpoints at P and D — even though you don’t see it. (That may seem a bit weird, but this idea’s just one of the rules of the geometry game. Don’t sweat it.)