# What Are Coplanar Points? - Definition & Examples

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• 0:01 Definition
• 2:21 Four or More Points
• 4:27 Importance of Coplanar Points
• 5:08 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Often

Elizabeth has taught high school math for over 10 years, and has a master's in secondary math education.

In math class, you heard about coplanar points. What are they, and how can you determine if points are coplanar or not? Do coplanar points have any use outside of geometry class? Read this lesson and find out!

## Coplanar Points: Definition

Coplanar points are three or more points which lie in the same plane. Recall that a plane is a flat surface which extends without end in all directions. It's usually shown in math textbooks as a 4-sided figure. You can see that points A, B, C and D are all coplanar points on a single plane:

The concept of coplanar points may seem simple, but sometimes the questions about it may become confusing. With a little bit of geometry knowledge and some real-world examples, you can master even the most challenging questions about coplanar points.

Any three points in 3-dimensional space determine a plane. This means that any group of three points determines a plane, even if all the points don't look like they're located on the same flat surface.

To see this, we're going to change our original picture slightly. In this new picture, the plane now has a line passing through it. The solid part of the line is above the plane, and the dashed part of the line is below the plane. You can make your own model at home by passing a knitting or sewing needle through an index card. We can see that points A, B, C and D are all still coplanar points. Point E is not coplanar with the original four points. But if I pick any group of three of the points, even a group containing point E, those three points will be coplanar. The reason is the statement given above - any three points in 3-dimensional space determine a plane.

Therefore, all of the following groups of points are coplanar:

• A, B, E
• B, C, E
• C, D, E

As you can see, you can use the three points to create a triangle. A triangle is a plane figure. Therefore, any set of three points is coplanar. You could even select a point in New York City, one in London and one in Mexico City, and the points would still be coplanar!

## Four or More Points

Once we get to four or more points, the situation changes. Groups of four or more points may be coplanar, or they may not be. Let's look at the picture again. A, B, C and D are still coplanar; however, A, B, C and E are not coplanar. This may be more obvious if we draw lines connecting the points. The black lines are in the plane, but the red lines are both above the plane. The 4-sided figure is not a flat plane, but is bent.

Let's take a look at a real-life situation, where the knitting needle has been passed through a piece of paper. The black yarn is on the paper, but the red yarn is above the paper.

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