Example Problems

Some examples:

In the circle below, if x is 120, what fraction of the area of the circle is the area of the shaded region?

Looking at our equality above, we can see that 

$$frac{120}{360}=frac13=frac{area;of;sector}{area;of;circle}$$

In this case, the area of the shaded region (sector) is one third the area of the entire circle.

In the circle below, if x is 110, minor arc MN is what fraction of the circle?

Going back to our equalities, we know the central angle, so we can plug that in.

$$frac{110}{360}=;frac{11}{36}=;frac{arc;length}{circumference;of;circle}$$

One more example:

In the circle below, if minor arc MN = (frac{2pi}5) , what fraction of the circle is arc MN?

When we see (pi) in our problem, we know we’re dealing with radians.  The formulas stay the same; we just have to remember it is (2π) all the way around the circle rather than 360°.

$$frac{frac{2pi}5}{2pi}=frac{2pi}5timesfrac1{2pi}=frac{2pi}{10pi}=frac15;frac{arc;length}{circumference;of;circle}$$