 Hi, this video is called Practice Problem 4. It is your last video of the day. The process of what you're doing using the 1 half per meter times apathem, I think you're getting fairly used to and comfortable with. But this one is definitely designed to help you practice with your rounding because it does say round to the nearest 10th. Sorry about that. So I was saying this exercise is going to be really help you practice with rounding and rounding correctly. So I urge you to punch everything in your calculator. Don't just take my word for it and you can see how you will round. The rule of thumb though is to keep things as exact as possible is never to round until the very, very, very end. And that is a difficult thing to do because you're going to be dealing with numbers that have lots of numbers past the decimal. They're going to go out very far and we're going to try really hard not to round it until the very end so we can get a good answer. All right, so let's just worry about that when we get there but know that it is coming. We're going to find the area of a regular octagon. An octagon has eight sides. It looks like we're given the perimeter which is 48. So let's think about this. A regular octagon, we get to use our formula of 1 half per meter times apathem. So all we have to do is figure out what our perimeter is and what our apathem is and we're set. We are already given a perimeter of 48 inches so I can plug that one in already. So the rest of our time is going to be devoted to figuring out what this apathem is and make sure we round correctly, okay? When we draw an octagon with eight sides, you can see I'm not very good at drawing them but I'm going to try my best and I want you to try your best now. So go ahead and draw a figure, regular octagon, so you're channeling a stop sign. So eight sides, I think it helps to go in and mark all eight sides as being congruent. It kind of helps your picture make a little more sense. If you want to even draw dots at the vertices to help your picture make a little more sense, you could certainly do that as well just to organize it a little bit. All right, so like all of these problems to start, draw in that central point and since an octagon has eight sides, we're going to draw eight radii. So we're going to create eight central angles. We're going to create eight triangles. Let's back up for just a second. Since the perimeter was 48 and there's eight sides, I know that all sides are six units long. Now let's go ahead and find that central angle. Since an octagon has eight sides, there's going to be eight central angles and 360 divided by 8 gives me 45. So I can see that my central angles are all 45 degrees. Now be careful. We're not going to end up using special right triangles because they're not 45, 45 triangles that I kind of was hoping they would be because when you drop down that opothum, the 45 gets cut in half and becomes 22.5. So that opothum bisects the 45 into 22.5. It creates a right angle right there and it splits the side length. It bisects the side length into three and three. So when I draw this little triangle right here, I've got a right angle, got a side length of three. Let's go ahead and call the opothum X. This angle up here was 22.5, which would make the bottom right hand angle be 67.5. How did I know that? I knew that because the 90, the 22.5 and the 67.5 all have to add up to 180. Okay? So for a fleeting second, I thought that I was maybe going to use 45, 45, 90 triangles, but now I realize that's not what I have. I've got a triangle that has angle measures of 22.5 and 67.5. So my only option here is Sokotoa and please remember, my goal, the whole reason I'm doing this is to find that opothum which I have here marked as an X. So if I can figure out what X is, I've got my opothum. I think I'm going to go ahead and go from the perspective of the 67.5, which makes the X the opposite, the 3 the adjacent, and opposite and adjacent is a tangent. So I've got a tangent of 67.5 equals the opposite, which is X over the adjacent, which is 3. I'm going to make the tangent of 67.5 into a fraction by putting it over 1. That allows me to cross multiply. X times 1 is X, and then I'm going to have to multiply the tangent of 67.5 times 3. The X is alone, so I'm allowed to multiply. Here's where the decimals come in and it's going to start getting tricky and we're not going to round for a long time. So please take really good notes here and practice with your calculator. So pick up your calculator right now and punch in the tangent of 67.5 times 3. I got a big long answer. It says 7.242640687. Your calculator might go out further than that and give you more numbers or maybe your calculator doesn't go out quite that far and doesn't give you quite as many as I got. The important thing is, is to leave that number showing on your screen, we cannot round yet. So you're going to keep as many numbers as possible. Okay? So basically since that was the epithome, we found that to be the epithome, it was right here in this little triangle. That was my epithome. I'm going to take this number and put it up here in my formula. Now I don't really feel like rewriting it. I'm safe because I still have that 7.24 long number in my calculator. So what I'm going to do is kind of work backwards. With this 7.242640687 in my calculator, I'm going to do times 48 and press equal. Then I have one half times, the big number I have now is 347.646753. Again, I'm not going to round to the very end and I'm lucky because I still just have this big number on the screen showing on the screen in my calculator. So I don't even really have to write it down at this point, but I do have to remember, I have to multiply by one half, which is the same thing as divide by two. So if you take that long answer and divide by two, you get your area to be 173.8233765. Now I am finally ready to round because I've arrived at the end. There's no more calculations to do. This is my area. So to round to the nearest tenth, it will round to 173.8 inches squared. Okay? So the key to this is not rounding to the very end and remembering to use your calculator. Keep the big number on the screen so you can just use it. If you do end up having to write it down and retype it in, I know that's annoying, but it only takes a minute and it's worth it to get the right answer because you rounded correctly.