It’s Professor Dave, let’s learn about perimeters.
Sometimes in movies, someone will shout: “secure the perimeter!”
What this means is to guard or enclose the boundary of an area so that no one gets in
or out.
In geometry, the word perimeter has a similar meaning, it’s the distance around a shape
or figure.
Finding the perimeter of a polygon is usually pretty easy, we just have to add up the lengths
of all the sides, and that will give us the distance around the shape.
This was one of the earliest mathematical necessities, as early farmers had to know
how much material they needed to enclose their land and guard their animals or crops.
Here is an irregular pentagon.
Given these sides lengths, we just add them up, and that’s the perimeter.
This is the same thing as asking: if you walked along each edge of the shape, all the way
back to where you started, how far did you walk?
If we don’t know one or more of the sides, we can usually calculate them, depending on
what shape we are looking at.
First let’s look at rectangles.
These have both a length and a width.
Since there are four sides total, two lengths and two widths, the perimeter of a rectangle
is equal to L plus W plus L plus W, or two L plus two W. So if the dimensions of a rectangle
are four by two, we can just plug these into the equation, and we see that we get twelve
for the perimeter.
For squares it’s even simpler, it’s just the side length times four, since all the
sides are the same.
Sometimes we may be given a perimeter problem that we can do algebra with.
Say we have a rectangle with a perimeter of twenty-four, and its length is three times
its width.
What are the dimensions of the rectangle?
Well we know that two L plus two W must equal twenty-four.
We also know that L equals three W. Remember how to solve two equations with two unknowns
by substitution?
Let’s put three W in place of L in this equation, which gives us six W plus two W,
or eight W, equals twenty-four.
We divide by eight, W must be three, and therefore L must be nine.
The rectangle is three by nine.
For a triangle, again we add up the sides, and if we have a right triangle with an unknown
length, we know that we can use the Pythagorean theorem to find the missing side.
Like this one, if the legs are three and four, we know the hypotenuse must be five, and the
perimeter is therefore twelve.
Sometimes we may be given an irregular polygon like this one, with some of the side lengths
filled in and others not.
This isn’t too hard once we realize that we can figure out the missing side lengths
by looking at the others, and imagining a rectangle behind this shape.
If these two sides add up to become one side of the imaginary rectangle, then these two
sides must add up the same value, so we just find the difference to label the missing side.
We can do this for all the missing sides, and then we just add up all the side lengths
to get the perimeter.
So that’s all we need to know to find the perimeter of any polygon.
Remember, this works for shapes made of straight line segments only, when we introduce curvature,
things get much trickier very quickly, and if the curvature is not perfectly circular,
we will need calculus to do it.
We aren’t there yet, but speaking of curvature, we haven’t discussed circles very much up
to this point, so let’s move on to those next, right after we check comprehension.