00:01
Okay, so for this problem, we're given a polygon that we can identify as a heptagon because it has seven sides.
00:11
And we can see on this diagram that the apathem of this polygon is equal to 5 centimeters.
00:18
And we're supposed to use this information to figure out the area of this polygon.
00:23
So we know that the area of polygon, of any polygon, is equal to one half the apathem, the perimeter and since we have the apthem all we need to do is find the perimeter we know that the perimeter of a polygon is equal to the number of sides times the length of each side gel -breathy is s so the way that we're going to figure out s just given the aphthm is by doing some trig so the first thing we need to figure out is what is the central angle of a heptagon well we know that the central angle, which i'll abbreviate ca, of any polygon is equal to 360 degrees, divide upon n, which is the number of sides.
01:18
In this case, it's seven.
01:21
And because this is not an exact number, we can just rank the 10th on paper, but when you're doing calculations, you should always keep the exact number in your calculator while you're still, you know, multiplying and adding to this number like we'll do later.
01:42
So again, on paper, i'm going to round to the nearest tenth.
01:46
I'm going to say this is 43 .7 degrees.
01:53
But in your calculator, you should keep that exact number.
01:56
All right.
01:57
So now that we know the central angle, we can kind of visualize this.
02:03
Like if we, we can say that this right here is going to be the center point of the heptagon.
02:09
Now, on a heptagon, there's also, like, adjacent vertices.
02:18
So let's say we have two points, and each one of these new points is a vertex.
02:25
And these vertices are adjacent to each other.
02:28
So if i draw a line from one to the other, this is actually a side length.
02:32
This is a side of the heptagon.
02:37
So it's going to be equal to s, which is what we're trying to find.
02:40
And then if we draw a line from each vertex to the center point, which i'll do right here, we create this triangle in which this angle right here is equal to the central angle.
03:00
And this is helpful because what we're going to do next is we're going to drop an apatham.
03:06
From the center point down to the midpoint of the side length.
03:14
It's going to make a 90 degree angle with the side, and it's going to bisect the central angle...