A circle comprises , so a sector comprising  of the circle will have an angle that is  of .

Therefore:

A circle comprises , so a sector comprising  of the circle will have an angle that is  of .

Therefore:

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  of the circle because,

 .

Since a circle has , the angle of the sector is:

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 

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

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

To find the percentage of the area of the circle that sector  covers, divide the area of sector  by the total area of the circle:

Area of the circle:

Percentage:

To go from a decimal to a percent, multiply by . This gets us to , 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:

From here multiply by 100 to get the percentage.

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, . The given sector has an angle of , so whatever percent this is of  will tell us what percent of the circle's area is covered by the sector:

The full measure of a circle is , so any sector will cover whatever fraction of the circle that its angle is of . We are given a sector with an angle of , so this sector will cover a percentage of the circle equal to whatever fraction  is of . This gives us:

The full measure of a circle is , so any sector will cover whatever fraction of the circle that its angle is of . We are given a sector with an angle of , so this sector will cover a percentage of the circle equal to whatever fraction  is of . This gives us: