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What is a Plane in Geometry? - Definition & Examples

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  • 0:00 What Is a Plane?
  • 1:08 Drawing and Naming Planes
  • 2:00 Examples
  • 3:45 Lesson Summary
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Lesson Transcript
Instructor
Miriam Snare

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

This lesson will help you understand the geometry concept of a plane. We will discuss how to name planes and look at some example problems. Then, you can check your understanding with a quiz.

What Is a Plane?

In geometry, a plane is a flat surface that extends forever in two dimensions, but has no thickness. It's a bit difficult to visualize a plane because in real life, there is nothing that we can use as a true example of a geometric plane. However, we can use the surface of a wall, the floor, or even a piece of paper to represent a part of a geometric plane. You just have to remember that unlike the real-world parts of planes, geometric planes have no edge to them.

In algebra, we graph points in the coordinate plane, which is an example of a geometric plane. The coordinate plane has a number line that extends left to right indefinitely and another one that extends up and down indefinitely. You can never see the entire coordinate plane. The fact that it extends forever along the x- and y-axis is just indicated by arrows on the ends of the number lines. Those are the two dimensions over which a plane extends forever. When you graph points, you never graph one point deeper into the paper than another point. That shows that the coordinate plane does not have thickness to it.

Drawing and Naming Planes

In order for us to discuss planes, we need to be able to see them and label them. Therefore, even though geometric planes do not have to edges to them, when they are drawn, they have an outline. Usually, they are represented by a parallelogram that is shaded in, like this:

Plane

If we want to talk about two or more different planes, then we need to be able to name each plane. There are two ways to label planes. Most frequently, you use three or four of the points that are in the plane as the name. Remember that points are indicated with a dot and are labeled with a capital letter. The second way to name a plane is with just one capital letter that is written in the corner of the image of the plane. This letter does not have a dot next to it and is sometimes written in a script font that is different from the font used for points.

Now let's go through some example questions to help you understand which points are in a plane and how to use them to name the plane.

Examples

Look at this image:

Plane

Now think about the answers to these three questions, and I'll explain the answers shortly.

  1. What points are in the plane?
  2. What point is not in the plane?
  3. What are three different names for the plane?

Now, let's talk about the answers.

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Additional Activities

Additional Examples and Discussion

In the following examples, students will demonstrate their understanding of planes, including which points are in the plane and which are not, as well as proper plane naming conventions. After completing the examples, students will have a solid understanding of the basics of planes in geometry and will be ready to move on to the discussion. In the discussion, students will be challenged to think of more than one plane at a time - imagining possible intersections of these flat, infinite objects in geometry.

Examples

Use the following image for examples 1 - 3.

1. What points are in the plane?

2. What points are not in the plane?

3. Which of the following would not be an appropriate name for the plane? B , ebf, fec, face, ebfc

Use the following image for examples 4 - 6.

4. What points are in the plane?

5. What points are not in the plane?

6. List all possible names for the plane.

Solutions

1. Points e, b, f, and c are all in the plane. Point e looks like it may be on an edge, but since planes extend infinitely, it is actually entirely within the plane.

2. Points a and d are not in the plane. They can be viewed as either floating above the plane in space or below the plane in space.

3. The name "face" would not be appropriate for this plane, because the point a is not inside the plane. All of the other names are appropriate, because they either consist of 3 or 4 points in the plane or is the capital letter B, which is in a different font and not next to a dot, and so is not referring to a point.

4. Points x, y, and z are all in the plane. Points y and z appear to be on an edge, but since planes extend infinitely, they are both actually entirely within the plane.

5. Points u, v, and w are all not in the plane. They are either above or below the plane in space.

6. One possible name is P, since it is in a different font and not next to a dot and so is not referring to a point. Other possibilities would all involve 3 or 4 points in the plane. We only have 3 points labeled in the plane, so the only other possibilities are all of the ways to order these points: xyz, xzy, yxz, yzx, zxy, or zyx.

Discussion

Imagine two different planes in three dimensional space. If the planes intersect each other, how do they intersect? Do they only touch in one point? A few points? Infinitely many points?

Guide to Discussion

It may help students visualize planes intersecting if they have paper to use as a prop. Two sheets of paper can be used to represent the planes - but students need to remember that planes extend infinitely - so there are no edges to the planes. If students believe that the planes only touch in one point, remind them of how the planes extend forever. The goal is to have students discover that there are two options for how the planes intersect - either they are directly on top of each other and so intersect everywhere (and in fact are the same plane) or they intersect in a line. The only other possibility is that the planes do not intersect - this is when they are parallel.

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