 Hi, this video is called Find the Exact and Approximate Area of the Shaded Region 1. So this is just like the ones you've been doing where we're trying to find the area of the shaded region, but they want two answers. They want an exact answer and an approximate. The approximate will have decimals because we'll be doing some rounding. The exact answer won't have any decimals. So we'll keep things in terms of pi, square roots, different things like that. So let's see where this problem leads us. The approach is the same. To find the area of the shaded region, imagine that the bigger shape, in this case it's a circle, is a piece of construction paper and someone cut out the triangle so you can see through it now. So what's left is the shaded region, so we're going to do area of the circle minus area of the triangle. So we have two different pieces that we have to solve and once we do that, we'll be all set. It looks like to me, area of a circle, the formula we'll use is pi r squared and then for area of the triangle, we have many options for area of a triangle. I don't know really what quite approach would be the easiest. I'm going to try 1 half perimeter times the opothum today. So let's see what happens. Look at this picture. The one big important piece of information we're given is that this right here is 8. I look at that and it's a radius of the triangle but it's also the radius of the circle because it goes from the middle of the circle to the edge on the circle. So that makes finding the area of a circle a piece of cake. The radius is 8, we will get area of a circle to be 64 pi. So I can fill that in and I'm already halfway there. Now I just have to concentrate on finding the area of my triangle using 1 half perimeter times the opothum. So if I can figure out what my perimeter is and my opothum is, I will be all set. I do think I'm going to have to do a little bit of work to figure this out. So let's go back to the picture. One radius of 8 was drawn in. Since a triangle has three sides, we can split it into three. If we do 360 divided by 3, that gives you 120 degrees. So my three central angles are all 120. But before I write the 120 in on that bottom triangle, I'm going to drop down an opothum and divide the 120 by 2. So that will give me 60 and 60 with 90 degree angles. This is worth taking, I'm going to take this little guy right here. I've got my 90 degree, my 60 degree, my 30, and I have this radius marked as an 8. I like this because if I can figure out what this bottom side is, that will help me get the perimeter, so that will help me get this part of my formula. And if I can figure out what this is, that's my opothum and that will help me get the second part of my triangle formula. So I like this approach. Okay. In my 30, 60, 90 opposite the 30 is n, opposite the 60 is n root 3, and opposite the 90 is 2n. Well, if 8 equals 2n, then n is going to equal 4. So that gives me this side is 4 and this side is 4 root 3, and I am feeling pretty good right now because I just found my opothum to be 4, that was right here, so the 4 can go into my formula. And now the perimeter, I think I have it. If this is 4 root 3, that means this is 4 root 3, so that entire bottom side of my triangle is 8 root 3. All sides of my triangle are 8 root 3. So to get the perimeter, I will do 8 root 3 times 3, which gives me 24 root 3. So now I'm ready to simplify. Area of a triangle is 1 half times 24 root 3 times 4. Since those three things, I'm all multiplying together, I can do it in whatever order I want. I'm going to go left to right. 1 half times 24 root 3 is the same thing as 24 root 3 divided by 2, so that gives me 12 root 3 times 4. Then I can multiply the 12 and the 4 together to give me 48 root 3 is the area of my triangle. So I will plug that into my original formula. Area of the triangle is 48 root 3. So now that is actually the exact answer because the 64 pi and the 48 root 3, they're not like terms and there's a subtraction sign between them so the rules are strict, so I can't combine them anymore. All I can do is put a label on it. It will be units squared. So this right here turned out to be my exact answer. Now to do the approximate, we're going to do some rounding. So we're actually going to punch 64 times pi in our calculator. We're going to punch 48 root 3 in our calculator and see what we get. When you punch in 64 pi, I got it to be 201.06192. And then when I punched in 48 root 3, I got 83.138438. It's too early at this point to round so I keep those long numbers. You might have to write them down. You punch them in your calculator and you should get 117.92349. So now if I'm going to round that to the nearest 10th, I look at the 9, I look at the number after it since the 2 is not 5 or bigger. It's going to stay 117.9 units squared and that will be my approximate answer because I did some rounding. So exact is when you keep it in terms of pi in the square root, the approximate is when you go ahead and figure out the decimals, just don't round until the end.