# Graphing Polygons on the Coordinate Plane

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will learn how to graph a polygon on a coordinate plane given its vertices as points. We will also look at different characteristics of the polygon that we can find by graphing the polygon.

## Graphing Polygons on the Coordinate Plane

Let's play a game! I'm going to give you some points that I want you to plot on the coordinate plane. Quick reminder: The coordinate plane consists of two number lines drawn perpendicular to one another, intersecting at 0 on both lines. After plotting the points, connect the dots using straight lines, but don't let any lines cross. Then, I want you to tell me what shape you end up with. Okay, here goes! Your points are (1,1), (3,4), and (5,1).

Let's see what you should end up with! First, we plot the points.

Next, we connect (1,1) and (3,4) with a line, then do the same for (3,4) and (5,1). Lastly, we connect the last point, (5,1), with the first point (1,1).

Ta-da! We have a triangle! Is this what you got? If so, you won the game! Not only that, but we also just performed a mathematical feat! We just graphed a polygon on the coordinate plane.

A polygon is a two-dimensional shape with three or more straight edges. The points at which the edges of a polygon meet are called vertices. When we are given the vertices of a polygon as points, we can graph the polygon on a coordinate plane. So really, this process is as simple as playing a game! The steps we take to graph a polygon on the coordinate plane are as follows:

1. Plot the given vertices on the coordinate plane.
2. Connect the vertices using straight lines, taking care not to cross any lines, forming a polygon.

The only thing to keep in mind is that when you connect the vertices, the lines you use must not cross. This will ensure that you create a polygon and not just a bunch of random lines between points.

## Applications of Graphs of Polygons

This game of graphing polygons is pretty fun and all, but why do we care to do this? Graphing polygons on the coordinate plane does more than just allow us to identify a polygon. It also allows us to find different characteristics of the polygon. For example, consider our triangle and suppose we want to know the distance of the side of the triangle that we drew from point (1,1) to point (5,1).

You'll notice that this side runs from point (1,1) and (5,1), and that these two points fall on the same horizontal line. Based on this, it is easy to see that the length of the side is the distance from x = 1 to x = 5, or 4 units. In general, when two vertices fall on the same horizontal line, we can find the distance of the edge between them by finding the difference of the x-coordinates.

5 - 1 = 4

Similarly, if two vertices fall on the same vertical line, we can find the length of the edge between them by finding the difference between the y-coordinates. Therefore, if we graph a polygon on a coordinate plane, we can easily find the length of a side that falls on the same horizontal or vertical line.

Another useful application that can come from graphing a polygon is finding the area of a polygon. The area of a polygon is how much space is inside the polygon. When we graph a polygon on the coordinate plane, the area is equal to how many unit squares are inside the polygon, where a unit square is a square with side lengths of one unit.

This is all wonderful information, but personally, I learn better through actual examples, so let's take a look at one.

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