Area of composite figure with circles

- [Instructor] We're told the following figure is composed of a semicircle, right over here, a triangle, and a quarter circle. So this is a triangle, this is the quarter circle. What is the area of the shape? Round to the nearest hundredth. So, like always, pause this video and see if you can work through this on your own before we do this together. All right, now let's do this together. Let's go piece by piece. So I'm gonna start first with this semicircle, right over here. We know that the area of a circle is pi times the radius squared, but that would be the area of the whole circle. But we just wanna take half of that. So let's take pi times, what's the radius? We know the diameter is 13, so the radius is going to be 6.5. That distance is 6.5, so times 6.5 squared. Now this expression would be the area of the entire circle if we just kept going, but that's not what we're dealing with. We're dealing with half of the circle. So that's the semicircle right over there. Now let's think about, actually let's go to this quarter circle next. So this is this part right over here. And this is a quarter of a circle. What looks like it has a radius of 5. So this is, if I took the whole circle, its area would be pi times the radius squared. But this is one fourth of that. So we're going to divide that by 4. And then last but not least, let's think about the area of this, what looks like a right triangle. We know the height here, but what's this length over here? That's not gonna be the 6.5. Well, to figure that out, and let me get rid of the 6.5, so we don't confuse ourselves. To figure out this length, we just have to remind ourselves that the entire diameter of the big circle is 13. And we know that this length right over here is 5, 'cause this was a quarter circle. So the radius is gonna stay constant at 5. And so if we take the 13 minus this 5 right over here, this length, 13 minus 5 is 8. So the area of this right triangle is going to be this base, 8, times the height, times 5. That would give us the area of this entire rectangle. But of course this is a triangle. So we will then divide that by two. And so now we just have to get our calculator out and round to the nearest hundredth. So let me do that. I have 6.5 squared divided by 2 is equal to that times pi, is equal to this business. And now I'm going to say plus, and I'm gonna open a parentheses here, and I'm gonna do all of this business. So 5 squared is 25. 25 divided by 4, times pi, times pi, close parentheses, is going to be equal to all of this. And now we have to add to that, 8 times 5 divided by 2. 8 times 5 is 40, divided by 2 is 20. So we'll say plus 20. And we get this. And if we're rounding to the nearest hundredth, this is actually the hundredth place, hundredths place is right over here. This number, the thousandths place, is less than 5. So it would actually be 106.00 square units. So this would be 106.00, let's call it units. Units squared. And we are done.