40 Perimeter and Area
Perimeter
Perimeter is a one-dimensional measurement that is taken around the outside of a closed geometric shape. Let’s start our discussion of the concept of perimeter with an example.
Guided Example
Joseph does not own a car so must ride the bus or walk everywhere he goes. On Mondays, he must get to school, to work, and back home again. His route is pictured in figure 1.
The obvious question to ask in this situation is, “how many miles does Joseph travel on Mondays”? To compute, we each distance: 3 + 6 + 6 = 15.
Joseph travels 15 miles on Mondays.
Another way to work with this situation is to draw a shape that represents Joseph’s travel route and is labeled with the distance from one spot to another.
Notice that the shape made by Joseph’s route is that of a closed geometric figure with three sides (a triangle) (see figure 2). What we can ask about this shape is, “what is the perimeter of the triangle”?
Perimeter means “distance around a closed figure or shape” and to compute we add each length: 3 + 6 + 6 = 15
Our conclusion is the same as above: Joseph travels 15 miles on Mondays.
However, what we did was model the situation with a geometric shape and then apply a specific geometric concept (perimeter) to computer how far Joseph traveled.
Notes on Perimeter
- Perimeter is a one-dimensional measurement that represents the distance around a closed geometric figure or shape (no gaps).
- To find perimeter, add the lengths of each side of the shape.
- If there are units, include units in your final result. Units will always be of single dimension (i.e. feet, inches, yards, centimeters, etc…)
To compute perimeter, our shapes must be closed. Figure 3 shows the difference between a closed figure and an open figure.
Example 1
Find the perimeter for each of the shapes below.
- Add the lengths of each side.
- Sometimes you have to make assumptions if lengths are not labeled.
Solutions
- 12 units
- 40 feet
Example 2
How do we find the perimeter of this more complicated shape?
Solution
Just keep adding those side lengths. 6 + 7 + 4 + 4 + 5 + 6 + 2 = 34 units
If you look closely at the shapes in the previous examples, you might notice some ways to write each perimeter as a more explicit formula. See if the results from what we have done so far match the formulas below.
| Shape | Perimeter | |
|---|---|---|
| Triangle with side lengths, a, b, c: | [latex]P=a+b+c\\[/latex] | |
| Square with side length a: | [latex]P=a+a+a+a\\[/latex] [latex]P=4a\\[/latex] |
|
| Rectangle with side lengths a, b: | [latex]P=a+b+a+b\\[/latex] [latex]P=a+a+b+b\\[/latex] [latex]P=2a+2b\\[/latex] |
Circumference
You may realize that we have not yet discussed the distance around a very important geometric shape: a circle! The distance around a circle has a special name called the circumference. To find the circumference of a circle, we use this formula: C = 2πr
In this formula, π is pronounced “pi,” and is defined as the circumference of a circle divided by its