Finding perimeter of shape with arcs

0:00- [Presenter] We are told the arc with a radius of 4 centimeters has a central angle of 60 degrees. The arc with a radius of 7 centimeters has a central angle of 300 degrees. Find the exact perimeter of the shape. So, pause this video. Have a go at this before we do this together.

0:16Alright, now this is a fun problem. The perimeter is essentially all of this black that we see over here. So there's a couple of parts that we can get right out of the gate. We could figure out this length because we know this entire radius of the larger circle is 7 centimeters while the smaller circle has a 4 centimeter radius.

0:36So this is going to be the difference. This is gonna be 7 centimeters minus this 4 centimeters. So this is going to be 3, we'll assume everything's in centimeters, so I just don't have to keep writing the centimeters over and over again. Actually, I'll just write it this time.

0:49And, by the same logic, this length right over here is also 3 centimeters. So now we just have to figure out the two arc lengths. So let's just do this one first, and then we'll do this larger one. So we know that the circumference of a circle is 2 times pi times the radius, or we could say it's equal to the diameter times the radius.

1:08Now what's going to be the circumference if we were to say this whole circle, like I'm not drawing it that well, with the radius of 4 centimeters? Well then the circumference is going to be 2 times pi times 4. We're assuming everything's in centimeters. So it'd be 8 pi centimeters if we were to go all the way around.

1:27Now we're not going all the way around. This arc has a central angle of only 60 degrees, 60 out of a possible 360 if we were to go all the way around. So this arc length right over here, it's 60 out of 360 degrees times the circumference of this entire circle if we were to go all the way around.

1:48So what is this going to be? Well, 60/360 is the same thing as 1/6 times 8 pi, and so that's going to be equal to 8 pi/6, and they're both divisible, both 8 and 6 are divisible by 2, so this gets us 4 pi/3 centimeters. So this is 4 pi/3. Now we can use the same methodology to figure out this arc length.

2:16If we're thinking about the entire circle, including this part right over here, it would be 2 pi times this radius. So the whole thing would be 2 times pi times 7 centimeters, or we could say it would be 14 pi. But we're not worried about the whole thing. We're just worried about this arc that has a central angle of 300.

2:39So times 300/360 because that is the amount of the whole circumference that we're dealing with because there's 360 degrees if we were to go all the way around, but we are only dealing with 300 of those 360. Well, if we divide both 300 and 360 by 60, you get a 5 and a 6.

2:59So we get 14 pi times 5/6. And then, let's see, both 14 and 6 are divisible by 2. So we get 7 pi times 5/3, and so this is going to be 7 pi times 5 is equal to 35 pi/3. And so if we want to find the perimeter of everything we're talking about, we want to take this 35 pi/3.

3:30That's this whole part over here. We want to add that to the 4 pi/3. So plus 4 pi/3. And then plus these two 3 centimeters. So if we add those two, we'll just get 6. And so 35 pi/3 + 4 pi/3 is going to get us 39 pi/3 + 6. Well, what's 39 divided by three? It's 13.

3:54So this is all going to be equal to 13 pi + 6. That is the exact perimeter of the shape in centimeters, 13 pi + 6 centimeters.