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Partial circle area and arc length

Video transcript

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 for 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 2 so the area if we're talking about the whole circle it would be equal to pi times 2 squared pi times 2 squared 2 squared is of course 2 times 2 which is equal to 4 so our area is going to be equal to 4 times 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 this is going to be 1/2 this so set 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 going 3/4 of the way around the circle so this arc length is going to be 3/4 of the circumference so what is the circumference well we know the circumference is equal to 2 pi times the radius they tell us what the radius is it's equal to 4 so our circumference is equal to 2 pi times 4 let's see we can just change the order in which we multiply so it's 2 times 4 times 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 will care about this arc length it's going to be 3/4 times the circumference of the entire circle so 3 over 4 times 8 pi what is that going to be well what's 3/4 times 8 well 3 times 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 3/4 is going to be 6 or another way to think about it is 1/4 of 8 pi is 2 pi and so 3 of those is going to be equal to 6 pi so the arc length of the partial circle is 6 pi once again we knew that because those 3/4 of the way around the way that I knew it was 3/4 is that this is a 90-degree angle this is 90 degrees which is 1/4 of the way around a circle so the arc length that we care about is the 3/4 of our circumference