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Partial circle area and arc length

Video transcript

- [Instructor] Find the area of the semicircle. So pause this video and see if you can figure it out. So let's see. We know that the area of a circle is equal to pi times our radius squared. So, if we think about the entire circle, what is the area going to be? Well, they tell us what our radius is. Our radius is equal to two, so the area, if we're talking about the whole circle, it would be equal to pi times two squared. Pi times two squared. Two squared is of course two times two, which is equal to four, so our area is going to be equal to four times pi. Now, I wouldn't put four pi here, because that would be the entire circle. They want the area of just the semicircle, of just this region right over here. Well, the semicircle is half of the circle, so if I want the area of the semicircle, this is gonna be half this. So instead of four pi, it is going to be two pi square units. That's the area of the semicircle. Let's do another example. So here, instead of area, we're asked to find the arc length of the partial circle, and that's we have here in this bluish color right over here, find this arc length. And you can see this is going three fourths of the way around the circle, so this arc length is going to be three fourths of the circumference. So what is the circumference? Well, we know the circumference is equal to two pi times the radius. They tell us what the radius is. It's equal to four, so our circumference is equal to two pi times four. Let's see, we can just change the order in which we multiply so it's two times four times pi. This is going to be equal to eight pi. This is going to be equal to eight pi. Now, that is the circumference of the entire circle. If we care about this arc length, it's going to be three fourths times the circumference of the entire circle. So three over four times eight pi. What is that going to be? Well, what's three fourths times eight? Well, three times eight is 24 divided by four is six. So this is going to be equal to six pi. Another way to think about it, one fourth of eight is two, so three fourths is going to be six. Or another way to think about it is, one fourth of eight pi is two pi, and so three of those is going to be equal to six pi. So the arc length of the partial circle is six pi, and once again we knew that because it was three fourths of the way around. The way that I knew it was three fourths is that this is a 90 degree angle. This is 90 degrees, which is one fourth of the way around a circle, so the arc length that we care about is the three fourths of our circumference.