 In this video for practice problem 1, I'm asked to find the exact surface area of a regular hexagonal pyramid. So I'm going to start with the formula. We know that we're looking at surface area of a pyramid. So I'm going to write down the formula for that. And just to note, before we get started, they're asking us to find the exact surface area. That means that our answer is likely going to have pi or a square root in it. And since this figure doesn't have any curved edges, it's not going to have pi in it. So we can expect that we're going to have a radical in our final answer. So I'm going to make myself a little bit more room here. I have my formula written out and I also pulled aside an area of a figure of the base. And you'll see why that is in a minute. I like to write down all the values that I'm trying to find. The perimeter of the base, the slant height of the pyramid, and B represents the area of the base. And then we can go ahead and see what we have so far. Slant height is given to a slant height of 8, so we don't have to do any calculations for that. I'll put that in. And I'm told that the base edge length of my hexagon is 10. And so that means this value here is 10. And if it's a six-sided figure, a hexagon is a six-sided figure, to find the perimeter then it's just going to be 10 times the six sides. So I have my perimeter and I have my slant height. And now I need to do some work to find B, which is the area of the base. And when I'm asked to find the area of a regular polygon, I know that that's a problem where I'm going to use the formula with the apathome. And that's why I wrote down my hexagon here and I am going to need to write down that formula. So to find my B, the area of the base, that's going to be the formula one-half apathome times perimeter. That's where I'm going to make a right triangle to solve for the apathome of this hexagon. And so we always start that by finding our central angle and then creating the right triangle. This central angle here is found by dividing by how many central angles there are, how many sides there are of this hexagon. There's six sides and a hexagon. So my central angle here is 60 degrees. And remember then when I drop down to make my right triangle, we have to cut that 60 degrees in half. And I'm just going to bring my right triangle over here, make it a little bit bigger. So I know if the central angle is 60, that creates a 30 degree angle there and I have a 30, 60, 90 triangle. And let's put in the information that I know. I was told again that the base edge length was 10. That means this total length is 10. And so in my right triangle, I'm just going to cut that in half and I know that the short side of my right triangle is 5. And this is where I'm going to use a special right triangle rules to find this value of the apathome. Special right triangle rules, if this is my short side because it's opposite the 30 degree angle, this is N and I know some of you guys use X or 1, N, 2N for the hypotenuse, and then the apathome side is actually N root 3. And I don't need to do much work here because I'm told that N equals 5. So that means my apathome, when I plug in 5, is going to be 5 root 3. And so for my formula down here to find the area of the hexagon, I know that my apathome is 5 root 3. I already found out that the perimeter of my hexagon is going to be the side length times 6. 10 times 6 we said was 60. And so now I can plug these values in to find the area of that base. One half apathome, that's a 3 times the perimeter and now I need to simplify this. Remember we want this in exact form so I'm going to leave that root 3 alone and I'm just going to say one half times 5 times 60. And if you plug that all in your calculator you can just do 0.5 times 5 times 60. And that's going to give you 150 and then you bring that root 3 down. And this is going to be the area of my base, the area of the hexagon. And I can go ahead and put that number up here. And now we have all the values to plug into our formula. So now using these values I'm going to just rewrite the formula. The surface area is one half times my perimeter of 60 times my slant height of 8. That's a 60 by the way, doesn't look like a 6. Plus my B, my area of the base which is 150 root 3. And now when I simplify that I have two terms here. When I simplify this term again one half times 60 times 8 I can plug that in my calculator 0.5 times 60 times 8 and that's going to come out to 240. And then the second term 150 root 3 because I want an exact answer I'm just going to leave that as 150 root 3. And this is my final answer because I want it in exact form. I can't combine these two terms. And my final answer for the surface area is 240 plus 150 root 3 units squared.