An hour hand rotates 360 degrees for every 12 hours, so the hour hand moves \(\displaystyle \frac{360^{\circ}}{12\ hours}=30^{\circ}/hr\).

There are 4.5 hours between 3 PM and 7:30 PM, so the total degree measure is

\(\displaystyle 4.5\ hours*30^{\circ}/hr = 135^{\circ}\).

One full rotation of a circle is \(\displaystyle 360^{\circ}\), so if a sector covers \(\displaystyle 20\%\) of a circle, its angle will be \(\displaystyle 20\%\) of \(\displaystyle 360^{\circ}\). This gives us:

\(\displaystyle \angle=0.20(360^{\circ})=\frac{1}{5}(360^{\circ})=72^{\circ}\)

A circle comprises \(\displaystyle 360^\circ\), so a sector comprising \(\displaystyle 70\%\) of the circle will have an angle that is \(\displaystyle 70\%\) of \(\displaystyle 360\).

Therefore:

\(\displaystyle \angle=0.70(360)=\frac{7}{10}(360)=252^{\circ}\)

A circle comprises \(\displaystyle 360^\circ\), so a sector comprising \(\displaystyle 45\%\) of the circle will have an angle that is \(\displaystyle 45\%\) of \(\displaystyle 360\).

Therefore:

\(\displaystyle \angle=0.45(360)=\frac{9}{20}(360)=162^{\circ}\)

The hour hand moves around a circle from 3PM to 6PM. Since there are 12 hours on a clock and the hand is moving through 3 of them, the hand is moving through a sector comprising \(\displaystyle 25\%\) of the circle because,

 \(\displaystyle \frac{3}{12}=\frac{1}{4}=0.25\).

Since a circle has \(\displaystyle 360^\circ\), the angle of the sector is:

\(\displaystyle \angle=0.25(360^{\circ })=\frac{1}{4}(360^{\circ })=90^{\circ }\)

The town of Thomasville organized a search party to look for a missing chicken. The party consisted of groups of people choosing a sector and searching outward from the center of town. Find the angle for the sector of searched by each group if each group chose an equal sized sector, and there were 120 groups. 

Begin by dissecting the question and figurign out exactly what they are asking and telling you. It's a bit wordy, but what we are looking for is the measure of the central angle for each of the search-sectors

We are told that there are 120 equal sectors. 

We also know that a circle is made up of \(\displaystyle 360^{\circ}\)

So, to find the central angle of each sector, simply do the following calculation:

\(\displaystyle \frac{360^{\circ}}{120}=3^{\circ}/sector\)

The entire circle is 360 degrees, therefore we can set up proportions and cross multiply.

\(\displaystyle \frac{x}{100}=\frac{45}{360}\)

\(\displaystyle 360x=4500\)

\(\displaystyle x=12.5\)

To find the percentage of the area of the circle that sector \(\displaystyle BOX\) covers, divide the area of sector \(\displaystyle BOX\) by the total area of the circle:

Area of the circle:

\(\displaystyle \small A=\pi r^2=\pi (15\:m)^2=225 \pi\:m^2\)

Percentage:

\(\displaystyle \frac{27 \pi \:m^2}{225 \pi \:m^2}=\frac{27}{225}=\frac{3}{25} =0.12\)

To go from a decimal to a percent, multiply by \(\displaystyle 100\). This gets us to \(\displaystyle 12\%\), the correct answer.

In this question, the slice of cake can be thought of as a sector. We are given that its central angle is 80 degrees and asked to find what percentage of the whole it represents. Straightforward division is all we need here. We are not give a radius or any way of finding one. All we need to find is the percentage of the whole. To do that, recall that a circle has 360 degrees total and compute the following:

\(\displaystyle \frac{80}{360}=\frac{8}{36}=\frac{2}{9}=.2\bar{2}\)

From here multiply by 100 to get the percentage.

\(\displaystyle 0.22\cdot 100=22.2\%\)

So the first slice represents about 22.2% of the total cake!

The percentage of a circle covered by a sector is equal to the angle of the sector divided by the full measure of the circle, \(\displaystyle 360^{\circ}\). The given sector has an angle of \(\displaystyle 18^{\circ}\), so whatever percent this is of \(\displaystyle 360^{\circ}\) will tell us what percent of the circle's area is covered by the sector:

\(\displaystyle \frac{18^{\circ}}{360^{\circ}}=\frac{1}{20}=0.05=5\%\)