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

\(\displaystyle C = 12 \pi + 18 \pi = 30 \pi\).

The white sector has degree measure

\(\displaystyle \frac{ 18 \pi }{ 30 \pi} \times 360 ^{\circ} = 216 ^{\circ}\).

The radius of the circle can be found using the circumference formula, setting \(\displaystyle C = 30 \pi\):

\(\displaystyle C = 2 \pi r\)

\(\displaystyle 30 \pi = 2 \pi r\)

\(\displaystyle r = 30 \pi \div 2 \pi\)

\(\displaystyle r = 15\)

The area of the white sector is therefore

\(\displaystyle A = \frac{216}{360 } \cdot \pi r^{2} = 0.60 \cdot \pi \cdot 15^{2} = 225 \cdot 0.60 \cdot \pi = 135 \pi\).

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

\(\displaystyle A = 24 \pi + 40 \pi = 64 \pi\).

The degree measure of the arc of the shaded sector is

\(\displaystyle \frac{24 \pi}{64 \pi} \times 360 ^{\circ } = 135 ^{\circ }\).

The radius can be found by solving for \(\displaystyle r\) and substituting \(\displaystyle A = 64 \pi\) in the area formula:

\(\displaystyle A = \pi r^{2}\)

\(\displaystyle 64 \pi = \pi r^{2}\)

\(\displaystyle r^{2} = 64\)

\(\displaystyle r = 8\)

The length of arc \(\displaystyle \widehat{MN}\) is 

\(\displaystyle \frac{135}{360} \cdot 2 \pi r = \frac{135}{360} \cdot 2 \cdot 8 \pi = 6\pi\).

There are a total of 360 degrees to a complete circle. The shaded sector has degree measure \(\displaystyle 80^{\circ }\), so the unshaded sector has degree measure

\(\displaystyle 360 ^{\circ }- 80^{\circ }= 2 80^{\circ }\)

Also, a sector of \(\displaystyle N^{\circ }\) is \(\displaystyle \frac{N}{360} \times 100 \%\) of the circle, so, setting \(\displaystyle N = 280\), we find that the unshaded sector is 

\(\displaystyle \frac{280}{360} \times 100 \%\)

\(\displaystyle = \frac{28,000}{360} \%\)

of the circle. This reduces to

\(\displaystyle \frac{28,000\div 40 }{360 \div 40 } \%\)

\(\displaystyle =\frac{700 }{9 } \%\)

\(\displaystyle =77\frac{7}{9 } \%\),

the correct percentage.

There are a total of \(\displaystyle 360^{\circ }\) in a circle. The unshaded portion of the circle represents a \(\displaystyle 140^{\circ }\) sector, so the shaded portion represents a sector of measure

\(\displaystyle 360^{\circ } - 140^{\circ } = 220 ^{\circ }\).

This sector represents 

\(\displaystyle \frac{ 220 ^{\circ }}{360^{\circ } } \times 100 \%\)

\(\displaystyle = \frac{ 22000}{360 } \%\)

\(\displaystyle = \frac{ 22000 \div 40 }{360 \div 40 } \%\)

\(\displaystyle = \frac{ 550 }{9 } \%\)

\(\displaystyle = 61\frac{1}{9} \%\)

of the circle.

What percentage of a circle is covered by a sector with a central angle of \(\displaystyle 240^{\circ}\)?

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

\(\displaystyle \%=\frac{\theta}{360^{\circ}}*100\)

Where theta is our central angle.

Plug in our given degree measurement and simplify.

\(\displaystyle \%=\frac{240^{\circ}}{360^{\circ}}*100=66.67 \%\)

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.

\(\displaystyle \text{Arc length}=\frac{\text{central angle}}{360}(\text{circumference})\)

Plug in the given central angle and circumference to find the length of the minor arc \(\displaystyle AB\).

\(\displaystyle \text{Arc length}=\frac{40}{360}(81\pi)=9\pi\)

The radius \(\displaystyle r\) of a circle, given its area \(\displaystyle A\), can be found using the formula 

\(\displaystyle A = \pi r^{2}\)

Set \(\displaystyle A= 81 \pi\):

\(\displaystyle \pi r^{2} = 81 \pi\)

Find \(\displaystyle r\) by dividing both sides by \(\displaystyle \pi\) and then taking the square root of both sides:

\(\displaystyle \pi r^{2} \div \pi = 81 \pi \div \pi\)

\(\displaystyle r^{2} = 81\)

\(\displaystyle r = \sqrt{81} = 9\)

The circumference of this circle is found using the formula

\(\displaystyle C = 2 \pi r\):

\(\displaystyle C = 2 \pi (9)\)

\(\displaystyle C= 18 \pi\)

A \(\displaystyle 90 ^{\circ }\) arc of the circle is one fourth of the circle, so the length of the arc is 

\(\displaystyle L = \frac{1}{4}C\)

\(\displaystyle = \frac{1}{4} \cdot 18 \pi\)

\(\displaystyle = \frac{9}{2} \pi\)

If a sector covers \(\displaystyle \frac{4}{5}\) 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.

\(\displaystyle \measuredangle =\frac{4}{5}(360^{\circ})=288^{\circ}\)

So our answer is:

\(\displaystyle 288^{\circ}\)

The total number of degrees in a circle is \(\displaystyle 360^{\circ }\), so the shaded sector represents \(\displaystyle \frac{144^{\circ }}{360^{\circ }}\) of the circle. In terms of percent, this is 

\(\displaystyle \frac{144^{\circ }}{360^{\circ }} \times 100 \% = \frac{14,400^{\circ }}{360^{\circ }} \%= 40\%\).

The shaded sector is 40% if the circle, so the unshaded sector is \(\displaystyle 100 \% - 40\% = 60\%\) of the circle.