Sectors - GMAT Quantitative Reasoning

One full rotation of a circle is , so if a sector covers of a circle, its angle will be of . This gives us:

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:

, so, by the Multiplication Property of Inequality,

.

The degree measure of an arc is twice that of the inscribed angle that intercepts it, so the above can be rewritten as

.

Here we need to recall the total degree measure of a circle. A circle always has exactly degrees.

Knowing this, we need to utilize two other clues to find the degree measure of .

1. Angle measures degrees, because it is made up of line segment , which is a straight line.

2. Angle can be found by using the following equation. Because we are given the fractional value of its area, we can construct a ratio to solve for angle :

So, to find angle , we just need to subtract our other values from :

So, .

Call the length of a side of Square B . Its perimeter is , which is the radius of Circle A.

The area of the circle is ; that of the square is . Therefore, a sector of the circle with area will be of the circle, which is a sector of measure

This is the kind of question we can't get right if we don't know the trick. In a circle, the size of an angle at the center of the circle, formed by two segments intercepting an arc, is twice the size of the angle formed by two lines intercepting the same arc, provided one of these lines is the diameter of the circle. in other words, is twice .

Thus,

We can see that the points devide the of the circle in 5 equal portions.

The final answer is given simply by which is , this is the angle of a slice of a pizza cut in 5 parts if you will!

As we have seen previously, the 6 points divide the of the circle in 6 portion of same angle. Each portion form an angle of or 60 degrees. As we also have previously seen, the angle formed by the lines intercepting an arc is twice more at the center of the circle than at the intersection of the lines intercepting the same arc with the circle, provided one of these lines is the diameter. In other words, . Since is 60 degrees, than, must be 30 degrees, this is our final answer.

First we calculate the area of the pizza. The area of a circle is defined as . Since our diameter is 18 inches, our radius is 18/2 = 9 inches. So the total area of the pizza is square inches.

Since the sector of the pie he cut for himself is 36 degrees, we can set up a ratio to find how much of the pizza he cut for himself. Let x be the area of the pizza he cut for himself. Then we know,

Solving for x, we get x=25.45 square inches, which rounds down to 25.

To find the area of a sector, we need to know the total area as well as the fractional amount of the sector at which we are looking.

In this case, we find the total area by using the following equation:

Because line segment is our diameter, our radius is . Thus, our total area is:

We need to go one step further to find the area of sector . Simply multiply the total area by the fractional amount that sector covers. We are told it is of the circle's area, so do the following:

Thus, our answer is .

To find the area of a sector, simply multiply the total area of the circle by the fraction of the part you are looking at.

In this case, our area will come from the following:

To find the fractional part of the circle we care about, take the number of degrees in over the total number of degrees in a circle ():

So, we find our answer by multiplying these two parts together:

The area of a sector with a certain angle will be whatever fraction of the total circle's area the angle of the sector is of . This means we divide by , and then multiply that fraction by the total area of the circle to give us the area of the sector:

Assume Statement 1 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle is, and therefore, the central angle of the sector cannot be determined. Also, no information about the circle can be determined. A similar argument can be given for Statement 2 being insufficient.

Now assume both statements are true. Then the length of semicircle is equal to . The circumference is twice this, or . The radius can be calculated as , and the area, . Also, is of the circle, and the area of the sector can now be calculated as .

The degree measure of is half the degree measure of the arc it intercepts, which is . We can use the measures of the two given major arcs to find , then take half of this:

Between noon and 2:40 PM, two hours and forty minutes have elapsed, or, equivalently, two and two-thirds hours. This means that the minute hand has made revolutions.

In one revolution, the tip of an eight-foot minute hand moves feet, or inches.

After revolutions, the tip of the minute hand has moved inches.

To find arc length, we need to find the total circumference of the circle and then the fraction of the circle we are interested in. Our circumference of a circle formula is:

Where is our radius and is our diameter.

In this problem, our diameter is the length of , which is , so our total circumference is:

Now, to find the fraction of the circle we are interested in, we need to realize that angle is degrees. We know this because it is made by straight line . Armed with this knowledge, we can safely calculate the length of our arc using the following formula: