The circumference of the circle is the sum of the lengths of the arcs, which is

.

The white sector has degree measure

.

The radius of the circle can be found using the circumference formula, setting :

The area of the white sector is therefore

.

The area of the circle is the sum of the areas of the sectors, which is

.

The degree measure of the arc of the shaded sector is

.

The radius can be found by solving for  and substituting  in the area formula:

The length of arc  is 

.

There are a total of 360 degrees to a complete circle. The shaded sector has degree measure , so the unshaded sector has degree measure

Also, a sector of  is  of the circle, so, setting , we find that the unshaded sector is 

of the circle. This reduces to

,

the correct percentage.

There are a total of  in a circle. The unshaded portion of the circle represents a  sector, so the shaded portion represents a sector of measure

.

This sector represents 

of the circle.

What percentage of a circle is covered by a sector with a central angle of ?

To find the percentage of a circle from the central angle, we need to use the following formula:

Where theta is our central angle.

Plug in our given degree measurement and simplify.

So, our answer is  66.67%

Recall that the length of an arc is a proportion of the circumference, just like how the measure of a central angle is a proportion of the total number of degrees in a circle.

Thus, we can write the following equation to solve for arc length.

Plug in the given central angle and circumference to find the length of the minor arc .

The radius  of a circle, given its area , can be found using the formula 

Set :

Find  by dividing both sides by  and then taking the square root of both sides:

The circumference of this circle is found using the formula

:

A  arc of the circle is one fourth of the circle, so the length of the arc is 

If a sector covers  of the area of a given circle, what is the measure of that sector's central angle?

While this problem may seem to not have enough information to solve, we actually have everything we need.

To find the measure of a central angle, we don't need to know the actual area of the sector or the circle. Instead, we just need to know what fraction of the circle we are dealing with. In this case, we are told that the sector represents four fifths of the circle. 

All circles have 360 degrees, so if our sector is four fifths of that, we can find the answer via the following.

So our answer is:

The total number of degrees in a circle is , so the shaded sector represents  of the circle. In terms of percent, this is 

.

The shaded sector is 40% if the circle, so the unshaded sector is  of the circle.