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Finding the Perimeter & Area of a Polygon Graphed on a Coordinate Plane

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

So you've graphed a pentagon or an irregular octagon on a coordinate plane. How do you find its area? Its perimeter? In this lesson we'll discuss how to find the area and perimeter for any graphed polygon.

Definitions

Polygon shapes show up in man-made objects and crystalline structures all over our planet. Square ice cubes, rectangular walls, octagon stop signs, and similar shapes make up the civilized world. When we're trying to figure out how much carpet to buy, or how far it is around a football field, then area and perimeter calculations become important.

In this lesson, we will deal with the idea of finding area and perimeter when you don't know the dimensions of an object, just the locations of the corners on a coordinate plane. A few terms to review:

  • A polygon (from Greek polugonos, meaning many-angled) is a closed figure on a single plane (not three-dimensional) that is made up of straight sides and at least three angles.
  • Its perimeter is the distance around its edge, and its area is the measure of the surface inside it.
  • A coordinate plane is a flat surface (plane) that uses crossed number lines to establish coordinates (numbered locations) for each position on the plane.

Pulling all of this together, we can see that a polygon graphed on a coordinate plane would be a closed figure with straight sides drawn on a flat plane that has coordinates, or numbered locations, for every point on that figure. Because a polygon's sides are straight, we can identify its position on the plane using just its corners (each angle's vertex).

Five-sided polygon graphed on a coordinate plane
Polygon on Graph

Finding the Perimeter of a Graphed Polygon

The perimeter of any polygon may be found by adding up the lengths of its sides. For example, if you have a six-sided polygon with sides that are 1'', 3'', 5'', 4'', 3'', and 2'' long, you would add the numbers up. 1 + 3 + 5 + 4 + 3 + 2 = 18, so the perimeter of that hexagon would be 18 inches.

The thing about a graphed polygon is that you generally have to calculate the length of its sides to get the perimeter. Unlike the example, where we were given the length of each side, a graph merely tells you where the corners are.

So how do you calculate perimeter without the side lengths? Well, if a side is horizontal or vertical, it's simple arithmetic. Let's take a look at a graphed five-sided polygon shown.


Using the Pythagorean Theorem to calculate the perimeter
image Polygon Calculations


Notice that the two corners farthest to the right (6,5 and 6,-1) have the same x value. That means the length of that side will merely be the difference between the y values. From 5 to -1 is a length of 6, so we now know the length of one of the sides.

The others aren't so simple. How do we figure out a length if there is a difference in both x and y values? Well, the Pythagorean Theorem (in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides) gives us the key. Any graphed polygon edge may be expressed as the hypotenuse of a triangle. One leg will be the difference between the x values, while the other will be the difference between the y values. Using those two lengths, plus the Pythagorean Theorem, you can get the hypotenuse, which is the side length you're looking for.

Each of the sides in the example (other than the blue easy one) were calculated by building a right triangle that had the unknown side as its hypotenuse (the side opposite the right angle). By taking the square root of the sum of the squares of the other two sides of the triangle, you can calculate the hypotenuse--the length of the side you're looking for.

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