yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Partial circle area and arc length


2m read
·Nov 11, 2024

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.

More Articles

View All
Gradient and graphs
So here I’d like to talk about what the gradient means in the context of the graph of a function. In the last video, I defined the gradient, um, but let me just take a function here. The one that I have graphed is (x^2 + y^2) (f of xy = (x^2 + y^2)). So,…
Howard Marks: 5 Strategies to Outperform the Market in 2021
Or number five, you can try to look for exceptions. What I call special niches, special people, who hopefully can produce a good return with safety in a low return world. But those people are truly exceptional and not easy to find. What inning do you see…
Fentanyl Explained #shorts
Why does fentanyl feel so good? Let us try it so you don’t have to. Fentanyl reaches your brain in seconds, and like other opioids, binds to opioid receptors. It stops pain signals and also releases a flood of dopamine, so the pain melts away as you slide…
Animal Life in the Forest Canopy - Meet the Expert | National Geographic
And welcome back to the channel! We are live yet again for our fifth Meet the Expert. Oh boy, what a journey we have been on! We’ve been down deep into the ocean, we’ve met with experts who study bears, we’ve been out in Hungary to see venomous snakes, we…
Definite integral of piecewise function | AP Calculus AB | Khan Academy
So we have an f of x right over here, and it’s defined piecewise. For x less than zero, f of x is x plus one. For x greater than or equal to zero, f of x is cosine of pi x. We want to evaluate the definite integral from negative one to one of f of x dx. …
Ross Ice Shelf Research | Continent 7: Antarctica
The Ross Ice Shelf is the largest lump of floating ice in the world. So little is known; the surface of Pluto has much better imagery than what’s beneath Raphael. It’s a lot of new stuff that we’re going to discover. No one’s ever done this before; it’s e…