# GMAT Math : Circles

## Example Questions

### Example Question #10 : Dsq: Calculating The Angle Of A Sector

Above are the results of the election for student body president that Mike is about to publish in the school newspaper. At the last minute, his friend Veronica stops him and tells him that there was an error in one digit - Lealand got 181 votes, not 101 votes.

Mike's article includes a circle graph that he will now have to change to reflect this corrected information. By how many degrees will the angle measure of the sector representing Lealand increase (nearest whole degree)?

Explanation:

According to Mike's erroneous information, the number of people who voted was

,

101 of whom voted for Lealand. Therefore, Mike's initial circle graph would have a sector of degree measure

representing Lealand's share of the vote.

However, the corrected figures are

181 of which went to Lealand, so his sector will have measure

,

an increase of .

The correct response is .

### Example Question #1 : Dsq: Calculating The Length Of The Diameter

What is the circumference of a circle?

(1)The diameter of this circle is 10

(2)The area of this circle is

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

Explanation:

The calculation of the circumference is . From statement (1) we know that . Therefore , and we can calculate  using the formula. From statement (2) we know that . Therefore , and we can calculate  using the formula.

### Example Question #1 : Dsq: Calculating The Length Of The Diameter

If  and  are points on a plane and  lies inside the circle  with center  and radius 5, does  lie inside circle ?

(1) The length of line segment  is 7.

(2) The length of line segment  is 7.

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

Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

EACH statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are not sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.

Explanation:

(1) The max distance between two points in the circle is twice the length of the

raidus (diameter =  = ). However,  can still be anywhere on the plane

(outside of the circle) as the statement does not indicate otherwise. Therefore, this statement is insuffieicent.

(2) The length of the line segment from  to  is greater than the radius of the circle. Thus,  must be outside of the circle. This statement is sufficient.

### Example Question #1 : Diameter

Find the diameter of circle B.

I) Circle B has a circumference of .

II) Circle B has an area of .

Each statement alone is enough to solve the question.

Statement 1 is sufficient to solve the question, but statement 2 is not sufficient to solve the question.

Neither statement is sufficient to solve the question. More information is needed.

Statement 2 is sufficient to solve the question, but statement 1 is not sufficient to solve the question.

Both statements taken together are sufficient to solve the question.

Each statement alone is enough to solve the question.

Explanation:

We are given the area and circumference of a circle and asked to find the diameter.

Given the following equations:

We can see that having either area or circumference will allow us to find the radius and in turn the diameter.

Thus, either statement is sufficient by itself.

### Example Question #71 : Circles

Find the diameter of circle

I) The area of  of   is .

II) An arc making up  of  is .

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Both statements together are needed to answer the question.

Either statement alone is sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Either statement alone is sufficient to answer the question.

Explanation:

I) Gives us a portion of the area of the circle. From this we can find the total area and solve for the radius. Then we can double our answer to find the diameter.

II) Gives us a portion of the circumference. From this we can find the total circumference and work our way back to the radius and then the diameter.

### Example Question #1 : Dsq: Calculating The Length Of The Diameter

The circle of center  is inscribed in square . What is the diameter of the circle?

(1) The ratio of the diameter to the circumference of the circle is .

(2) The area of square  is .

Statments 1 and 2 together are not sufficient.

Each statement alone is sufficient.

Both statements together are sufficient.

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Statement 2 alone is sufficient.

Explanation:

To find the diameter of the circle we would need information about the square or about the circle itself.

Statement 1 gives us a ratio of the diameter to the circumference  of the circle.

If we write the equation it is  or .

Therefore, for all circles, the ratio of the diameter to the circumference will be .

This statement is not helping.

Statement 2 on the other hand gives us the area of the square and therefore allows us to calculate the side of the circle, which is the same as the diameter.

Hence, statement 2 alone is sufficient.

### Example Question #71 : Circles

Find circle  ratio of diameter to circumference.

I) The area of circle  is .

II) Chord  passes within  meter of the center and has a length of  meters.

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Explanation:

To find the ratio of diameter to circumference we need our diameter and circumference.

We can use I to find our radius and from there our diameter and circumference.

II can be used to make a triangle with sides of  which we can then use to find the radius and from there the diameter/circumference.

Either statement is sufficient to answer the question.

### Example Question #2 : Dsq: Calculating The Ratio Of Diameter And Circumference

What is the ratio of the circumference to the diameter of Circle ?

1.) The radius of the circle is .

2.) The area of the circle is

Both statements taken together are sufficient to solve the question.

Each statement alone is sufficient to solve the question.

Neither statement is sufficient to solve the question. More information is needed.

Statement 2 is sufficient to solve the question, but Statement 1 is not sufficient to solve the question.

Statement 1 is sufficient to solve the question, but Statement 2 is not sufficient to solve the question.

Each statement alone is sufficient to solve the question.

Explanation:

We are asked to find the ratio of the circumference  to the diameter  of Circle   and are given the radius and the area. We also know that .  Taking each statement individually:

1.) The radius  is  and we know that . The diameter . Since we can determine that ,  Statement 1 is sufficient to solve for the ratio by itself.

2.) The area  of the outside circle is , so therefore . Since we can use  to determine both the circumference  and diameter , Statement 2 is sufficient to solve for the ratio by itself.

### Example Question #1 : Dsq: Calculating The Ratio Of Diameter And Circumference

Circle A and circle B are given. If the diameter of circle B is , what is the diameter of circle A

1. The circumference of circle A is .
2. The ratio of the diameters of circle A and circle B is , respectively.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Each statement alone is sufficient to answer the question.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Each statement alone is sufficient to answer the question.

Explanation:

Statement 1: If we know the circumference, we can calculate the diameter.

If  then

Statement 2: We know the diameter of circle B is  and that the ratio of the circles. We can set up our proportions and find the diameter:

### Example Question #621 : Data Sufficiency Questions

What is the diameter of the circle?

1. The diameter to radius ratio is .
2. The circumference is .

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Each statement alone is sufficient to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Explanation:

Statement 1: We're given a ratio (which you should already know) but no values. We need additional information to answer the question.

Statement 2: If we're given the circumference, we can solve for the diameter.

which means

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.