 In number three we're going to continue to find the volume of a pyramid, but this one is a hexagonal pyramid. So that means that we're going to have to use the area of a regular polygon. So let's start with writing out volume is equal to one-third times the area of the base times the height. You should always start with writing out your formula. You want to make sure you pay attention to if you're doing pyramids or prisms and this one we're doing a pyramid. I'm going to take it to the next step and I'm going to say, okay, what is the area formula for my base? Since this base is a hexagon, we're going to use the area formula that is one-half times a pothem times perimeter and then we'll multiply that times the height of the pyramid. Now the height of the pyramid, that's simple. They tell us it's eight. So a little tricky is figuring out the area of the hexagon. So we are told that the base edge length is ten. So what I'm going to do on my picture is I'm just going to label that's ten. Well that leads us to be able to find the perimeter pretty easily. There are six sides in a hexagon, so ten times six is sixty. The problem is we need to find the apothem. So I'm going to, in my picture, draw the center of the hexagon. Here's the apothem and I'm going to make my right triangle just like we did in chapter 11. Label that A, the apothem, and because the side length is ten, I know that that side of the triangle is five. The triangle angle, 360 divided by six, and then divide that by two will give you the angle of this triangle, which is thirty degrees. So we know we're dealing with a thirty-sixty-ninety triangle. Therefore if this side of the triangle is five, then the apothem would have to equal five root three. So now I'm going to plug all of this into the formula, one-third times one-half times five root three times sixty times eight. Notice back in the directions that we want to keep it exact. When I multiply one-third times one-half, multiply fractions, it is one over six. And if I multiply five times sixty times eighty, we get twenty-four hundred root three. And so then if I do twenty-four hundred divided by six, I get four hundred root three. And that is cubic units. Number four is a regular triangular pyramid. When it is a regular triangle, that means that it's equilateral. So when I do the volume for this pyramid, it's going to be one-third times the area of the base times the height. And because this triangle is equilateral, I'm going to use my formula one-fourth times s squared, the side squared times root three times the height of the pyramid. And this turns into a fairly straightforward problem. We know that the base edge of the triangle is twelve and we know the height is ten. So if I plug those values in, one-third times one-fourth times twelve to the second power root three times ten. We know that twelve to the second power is a hundred and forty-four. So if we take the fractions, one-third times one-fourth is one-twelfth times a hundred and forty-four root three times ten. And then in your calculator, one-hundred and forty-four times ten is one-thousand four-hundred and forty. Divide that by twelve and it turns out you just get one-hundred and twenty. Don't forget to attach the root three and your cubic units. Now by the way, this one didn't ask us to give exact. I would just go ahead and leave it as exact. But if you wanted to enter this into your calculator and give the approximate answer, that would be okay because in this problem they don't tell you to either give exact or approximate. Number five is a little bit tricky so let's take a look at number five. Find the approximate volume of a regular pentagonal pyramid given an opothem of three and a slant height of five. So again I'm going to start my problem by writing out the volume formula, one-third times the area of the base times the height. In this case because we are working with a base that's a pentagon, we're going to have to do one-half times the opothem times perimeter. And then we're going to have to multiply by the height of the pyramid, which leads to our first issue. The first issue is we don't know the height. We know the slant height but we don't know the height. So take your picture and draw in the slant height and then also draw where your height would be and what you should see as a result is a right triangle. And if I know that this is five and if you can recognize that this right here is the same as the opothem, which is three, they told us the opothem is three, then we can use Pythagorean Theorem to find the height of this pyramid. So h squared plus three squared equals five squared and if you solve for h you should get a height of four. So we've got the height of the pyramid. The next problem we run into is we know the opothem but we don't know the perimeter of the pentagon. And so what we're going to do is we're going to draw, just like we would if we were finding the area of a pentagon, I am going to draw another right triangle but this time I'm going to draw from the center of the pentagon to the base and my opothem I know is three and what I'm going to do is I'm going to try to find this piece of the triangle. So we're going to have to involve the central angle. The central angle remember is 360 divided by 5 which is 72 degrees and then you have to take that in half, 72 divided by 2 to get this angle right here which is 36 degrees. So if I have a right triangle with an angle of 36 degrees in order to find x I'm going to have to use Sokotoa. So because I have x and 3 that's going to involve tangent so the tan of 36 equals opposite which is x over adjacent which is 3 and if I put that in my calculator 3 times the tangent of 36 degrees should be about 2.2. Now remember in the directions we are approximating so it's fine to round. Now if x is 2.2 I need to multiply that by 2 to get the side length so 2.2 times 2 is 4.4 and the whole reason for doing this is we had to figure out the perimeter of the pentagon so 5 sides times 4.4 is 22. So now I have the perimeter, I have the epithome of 3 and we have found the height which is 4 then we can finish the problem. One third times one half times the epithome of 3 times the perimeter of 22 times the height of 4. And if we plug this into our calculator it is going to give us an approximate answer of 44 cubic units.