 Hi, this video is called Practice Problem 1. I promise you, it will be a lot shorter than the video you just finished. But please, it's very important to go through these next four examples because the problems are all just a little different, and you will get good at these if you practice. So here, consider this. These next four videos, a chance to practice using what you've just learned. So this problem says find the exact area of a regular hexagon with a perimeter of 60 units. Well, we're told we have a regular hexagon. A hexagon has six sides. The picture proves that it has six sides. And since it's a regular hexagon, all six of those sides are the same. So since we have a regular hexagon to find the area, I'm going to use my formula of 1 half perimeter times a pothum. So my goal is to figure out what the perimeter is and what the pothum is, because once I have that, I'll just multiply the three things together. Well, the nice thing is that in the problem, they gave me one of the things I need. They told me the perimeter is 60. So I'll replace p with a 60. So this doesn't seem so bad. All I have left to do is figure out what the epothum is. Once I know that, I can just multiply everything together. One thing to note is that it says to find the exact area. That means no decimals are going to be allowed in your answer. So if you use your calculator, be careful. All right, let's get started. To find that epothum, when you do problems like this, always start by figuring out how many triangles you can make. A hexagon will have six sides, so it will have six radii. So it will make six triangles. All right? Let's think about what the length of these side lengths are. If the perimeter is 60, and there's six sides, 60 divided by 6 is going to be 10. End of the school day. 60 divided by 6 is going to be 10. So that's telling me that if the perimeter is 60, and all six sides are the same because it's regular, all of those sides will be 10 units long. So that's getting us closer to finding the epothum. Another thing to get us closer to finding the epothum will be to find the central angle. Remember the central angle is right here. Since there's a hexagon with six sides, there'll be six central angles. So we're going to do 360 divided by 6, which gives me 60 degrees. So all of these central angles will be 60. Now before I write the 60 in that bottom triangle, I want to think for a second. My goal is to find the epothum or the epothum. So let's drop that down. Well, what did the epothum do? It split my side length into two, five and five. And then it also split my 60 degree central angle into 30 and 30. So at this stage, it's a really good idea to take this little triangle and create it bigger so I can see what's going on. And one thing I forgot to mention, that epothum, it splits the segment. It bisects the segment to five and five. It bisects the central angle to 30 and 30. It also creates a 90 degree angle here. So that's nice. You've gotten a lot of practice with 90 degree angles. You know a lot about them. This is five. This is 30. Here's the epothum, which I'm looking for. What kind of triangle do I have? Well, if this is 30 and this is 90, this is 60. So you have to go back now and remember what you know about your 30, 60, 90 triangles. Opposite the 30 is n. Opposite the 60 is n root 3. And opposite the 90 is 2n. Well, this is pretty slick because n is five. So that would make my hypotenuse be 2 times 5, which is 10. And it would make the epothum, which is n root 3, it'll make it be 5 root 3. So which side do I care about for the terms of my problem? Can you remember in my problem, I'm looking for that epothum. And the epothum, we just found to be 5 root 3. So let's go ahead and replace the a with 5 root 3 in my formula. I'm feeling excited now because I can see I'm almost done because I filled in all the variables of my formula. All I have to do is solve. Since it is a multiplication problem, I can really multiply this in whatever order I want. I'm going to go from left to right. So first I'm going to do 1 half times 60. Multiplying by 1 half is the same thing as dividing by 2. So 1 half times 60 is 30. So then I have 30 times 5 root 3. Sorry, I don't know why it wrote like that. So this now, because it's multiplication, I don't need like terms. So I can go ahead and multiply the 30 and the 5 together because they're both outside of the square root. 30 times 5 is 150. I'll keep the root 3 there. And I'm going to put a label of units squared on my answer. Because it says find the exact area, I will leave my answer like this. I'm not going to do 150 times the square root of 3 in my calculator to see what decimal I get because my answer wouldn't be exact anymore, it would be approximated. So the area of that regular hexagon is 150 root 3 units squared.