# GMAT Math : Circles

## Example Questions

### Example Question #241 : Gmat Quantitative Reasoning

What percentage of a circle is a sector if the angle of the sector is ?

Explanation:

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:

### Example Question #12 : Geometry

What percentage of a circle's total area is covered by a sector with an angle of ?

Explanation:

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:

### Example Question #11 : Circles

A teacher buys a supersized pizza for his after-school club. The super-pizza has a diameter of 18 inches. If the teacher is able to perfectly cut from the center a 36 degree sector for himself, what is the area of his slice of pizza, rounded to the nearest square inch?

24

26

27

25

28

25

Explanation:

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.

### Example Question #12 : Circles

In the figure shown below, line segment  passes through the center of the circle and has a length of . Points , and  are on the circle. Sector  covers  of the total area of the circle. Answer the following questions regarding this shape.

Find the area of sector .

Explanation:

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:

### Example Question #13 : Circles

Consider the Circle :

(Figure not drawn to scale.)

If angle  is , what is the area of sector  in square meters?

Explanation:

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:

### Example Question #251 : Gmat Quantitative Reasoning

If a circle has an area of , what is the area of a sector with an angle of  ?

Explanation:

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:

### Example Question #15 : Circles

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: Arc  has length .

Statement 2: Arc  has length .

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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 .

### Example Question #1 : Calculating The Length Of An Arc

Note: Figure NOT drawn to scale

Refer to the above diagram.

What is  ?

Explanation:

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:

### Example Question #1 : Calculating The Length Of An Arc

A giant clock has a minute hand that is eight feet long. The time is now 2:40 PM. How far has the tip of the minute hand moved, in inches, between noon and now?

Explanation:

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.

### Example Question #3 : Calculating The Length Of An Arc

In the figure shown below, line segment  passes through the center of the circle and has a length of . Points , and  are on the circle. Sector  covers  of the total area of the circle. Answer the following questions regarding this shape.

What is the length of the arc formed by angle ?