00:01
Today, let's look at finding the volume of a regular pentagonal pyramid.
00:11
So let's set up the equation here.
00:13
Let's say we have our pentagonal pyramid here.
00:21
So i'll draw this as best as i can for you.
00:38
There we go.
00:46
There's our pentagonal pyramid.
00:48
Mid and we can go ahead and define some of these points here all right and let's say that the measure of angle a b c is equal to 35 degrees we can call the side length of the base three again don't forget that they are, it's a regular pentagon and so all the sides will be three.
01:25
Right.
01:28
And so there's our, there's our figure and there's some information.
01:30
So if i want to find the volume of this pentagonal pyramid, we're going to need a few formulas, right? and so first, we're going to need to be able to find the area of a regular polygon and to do so, right? that formula, if you remember, is one half.
01:52
Times the apothem times the perimeter and then the volume of a pyramid right we can find that volume using the formula one -third times the area of the base times the height and so there's a lot of moving parts that are going on with this problem and so first thing we want to do is find the apothem right and so to find the apothem again the the epithum is going to be this line, that red dotted line in our figure.
02:35
To do so though, we need to know a little more information.
02:40
And so if i were to draw that triangle, it would look something like this, okay? and we are trying to find this here, a.
02:53
All right.
02:54
We need that epatham.
02:56
Well, what are the other angles? and well, to do so, right, what's nice is if i were to circumscribe that pentagon, we would have a bunch of central angles in the middle.
03:08
And so since it's regular, i know all those angles should be the same.
03:11
And so, right, 360 degrees is a triangle.
03:15
If i divide that by five, it would give me 72.
03:18
However, the triangle that i'm looking at is only half that size.
03:23
And so half of 72 tells me that this top angle here has to be 36.
03:30
Degrees.
03:33
And so this is nice because we can actually use trigonometry to find this because the base of this triangle is part of the perimeter.
03:43
And so since the side length is three, i know half of that has to be 1 .5.
03:49
Right.
03:50
And so we can find, again, we can find the epotham by using trigonometry.
03:57
Right.
03:58
And so here, if i'm looking at this, i know.
04:03
That we are going to have to use tangent right because we're given the opposite and the adjacent side to the given angle to the reference angle so i know we're going to use a tangent of 36 degrees and that's going to be equal to 1 .5 divided by a right and so we're trying to solve for a we want to know what is the epotham and so if i rewrite this to solve for a we're going to get a is equal to 1 .5 divided by tangent of 36 which is going to be equal to approximately 2 .0646 right so there's our opatham and so now we can actually find the area of the pentagon right and so i know that the area of the pentagon it's going to be equal to one half times our potham 2 .0 646 times the perimeter again side length is three so three times five would give me 15 and so i know that the the area of our base is actually going to be 15 .4845 about all right so there is that.
05:51
However, that gives us, right, that is also equal to our capital b because the area is going to be equal to our area of the base.
06:03
And so this is also equal to our capital b.
06:07
And so we will need that a little bit later because now we need to know, right, to find the volume, now we need to know what the height is...