Find the area of the minor segment of a circle of radius 42cm, if length of the corresponding arc is 44cm.

924 cm^3

Step-by-step explanation:

With this kind of question it's important to know the area corresponds to how much of an arc is being made.  In other words, if you cut the circle in half, so the arc is half of the whole thing, the area will be half of the total area.  So we want to find four things.  Total area, circumference, arc length and are of the arc.  We'll go in that order.

Area of a circle is pi*r^2 so the radius is 42 cm which means the area is pi * 42^2 cm^2 = 1764*pi cm^2

Now the circumference, or in other words the arc length of the whole circle.  to find the circumference you use the formula 2*pi*r = 2*pi*42 = 84*pi cm

The question gives us the arc length we want to find the area of, so how much is it part of the whole circumference.  In other words, 44 is what fraction of 84*pi?  To hopefully make it more sense, 42*pi is half of 84*pi so that makes the ratio 1/2.  So to find the ratio of 44 is to use algebra, what fraction do we have to multiply 84*pi by to get 44.  so the equation is 84*pi*x = 44

84*pi*x = 44

x = 44/(84*pi)

x = 11/(21*pi)

To check let's go ahead and do the equation.

(84*pi)*11/(21*pi)

(84*pi*11)/(21*pi)

(84*1)/21

44

So just in case it's confusing 44 is 11/(21*pi) of 84, so the area of the section will be 11/(21*pi) of the total area, which is 1764*pi cm^3  So to find the area of the segment is gotten just by multiplying the total area by the fraction.

1764*pi * 11/(21*pi)

(1764*pi*11)/(21*pi)

19404/21

924 cm^3

Another way to do this is to set up a proportion, where you write the ratio of onearc and its area as a fraction equaling the same ratio of the other arc.  So there are two ways to do this.

(total area / total arc length[circumference]) = (segment's area/ segment's arc length) or (circumference / total area) = (segment's arc length / segment's area)  of course you are solving for the segment's areaso you will need to do some multiplying.

Let me know if any of this was confusing and I'd be happy to explain.