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Using Coordinates to Solve Perimeter & Area Problems

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  • 0:04 Coordinates
  • 1:43 Perimeter of Polygons
  • 4:25 Area of a Triangle
  • 6:35 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we'll show you how coordinates are used to calculate two important features of a plane figure. Using several examples, we'll demonstrate how to compute perimeters and areas using coordinates.

Coordinates

Fred is creating a fenced garden. He's decided where he can place each corner to have the biggest garden possible. Now he has to buy the fencing material and fertilizer for his plants. But how much of each does he need?

How can these corner locations help Fred determine the length of fencing he needs and the area in which he'll be spreading fertilizer? Well, he'll use coordinates. Coordinates give a point's location in the xy-plane. One special location is the vertex, the point where two lines of a shape form a corner. When we have the coordinates for all the vertices of a figure, we can calculate its perimeter and area.

Let's do an example using these four coordinates: (0,0), (0,3), (3,3), and (3,0).

Step 1

Plot the coordinates. The first number tells the x value while the second number is the y value. Thus, (3,0) is the point located at x = 3 and y = 0. Plotting the coordinates gives a picture of the figure.

Step 2

Connect the dots and identify the figure. The connected points show a square.


connect_the_dots


Step 3

Determine lengths. By counting the grid lines we see that the sides = 3.

Step 4

If a formula exists, use it to find the perimeter and the area. The perimeter can be calculated as P = 4s where s is the length of the side of the square. The area of a square A = s2. Thus,

P = 4s = 4(3) = 12

and

A = s2 = 32 = 9

If formulas for a figure do not exist, we can still find the perimeter by adding the lengths of the sides. Without an area formula, we could estimate the area by counting the number of little squares enclosed.

Perimeter of Polygons

A polygon is a shape having three or more sides. In this next example, we have 5 sides. We calculate the perimeter by adding the lengths of the sides. If the length is not obvious from the plot, use the distance formula which says the distance d between (x1,y1) and (x2,y2) is given by:


distance_equation


Let's look at an example in which we determine the perimeter for the polygon with these coordinates: (-5,5), (5,5), (7,-2), (1,-4) and (-7,0).

Step 1

Plot the coordinates.


plot_the_points


Step 2

Connect the dots and identify the figure. This is a 5-sided polygon.


connect_the_dots


Step 3

Determine lengths. It will help to label each of the five lengths:

  • d1 from (-5,5) to (5,5)
  • d2 from (5,5) to (7,-2)
  • d3 from (7,-2) to (1,-4)
  • d4 from (1,-4) to (-7,0)
  • d5 from (-7,0) to (-5,5)

We'll use the distance equation to find each length, although d1 can be found by looking at the graph. Five units to the left and five units to the right, gives d1 = 10. To check our results, we can use the distance formula for d1:


d1


We can find the other lengths the same way:


d2



d3



d4



d5


Note that we have kept two decimal places.

Step 4

If a formula exists, use it to find the perimeter. Recall that the perimeter (P) is the sum of the lengths of each side.

Thus, P = d1 + d2 + d3 + d4 + d5

P = 10 + 7.28 + 6.32 + 8.94 + 5.38

P ≅ 37.9

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