Partial circle area and arc length

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 * 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 were talking about the whole circle, it would be equal to Pi * 2^2.

2^2 is, of course, 2 * 2, which is equal to 4. So, our area is going to be equal to 4 * pi. Now, I wouldn't put 4 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, it's just going to be half this. So, instead of 4 Pi, it is going to be 2 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 what we have here in this bluish color right over here.

Find this arc length, and you can see this is 3/4s of the way around the circle. So, this arc length is going to be 3/4s of the circumference. So, what is the circumference? Well, we know circumference is equal to 2 pi * the radius. They tell us what the radius is; it's equal to four.

So, our circumference is equal to 2 pi * 4. Let's see, we can just change the order in which we multiply. So, it's 2 * 4 * pi. This is going to be equal to 8 Pi. This is going to be equal to 8 Pi.

Now that is the circumference of the entire circle. If we care about this arc length, it's going to be 3/4s times the circumference of the entire circle. So, 3/4 * 8 Pi, what is that going to be? Well, what's 3/4 * 8? Well, 3 * 8 is 24. Divided by 4 is 6. So, this is going to be equal to 6 Pi.

Another way to think about it, 1/4 of 8 is 2. So, 3/4s is going to be 6. Or another way to think about it is 1/4 of 8 Pi is 2 pi, and so three of those is going to be equal to 6 Pi.

So, the arc length of the partial circle is 6 pi, and once again we knew that because it was 3/4s of the way around. The way that I knew it was 3/4s is that this is a 90° angle. This is 90°, which is 1/4 of the way around a circle.

So, the arc length that we care about is the 3/4s of our circumference.