# GMAT Math : Circles

## Example Questions

### Example Question #3 : Dsq: Calculating The Area Of A Sector

The circle in the above diagram has center .  Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1:

Statement 2:

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

EITHER 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.

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.

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

Explanation:

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that .  is an inscribed angle that intercepts the arc ; therefore, the arc - and the central angle  that intercepts it - has twice its measure, or . From angle addition, this can be subtracted from  to yield the measure of central angle  of the shaded sector, which is . That makes that sector  of the circle. The white sector is  of the circle, and the ratio of the areas can be determined to be , or .

### Example Question #4 : Dsq: Calculating The Area Of A Sector

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

Statement 1: .

Statement 2: The circle has circumference .

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.

BOTH statements TOGETHER are insufficient to answer the question.

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

Explanation:

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle  of the sector.

Statement 1 alone gives us the circumference; this can be divided by  to yield radius , and that can be substituted for  in the formula  to find the area: .

However, it provides no clue that might yield .

From Statement 2 alone, we can find , an inscribed angle, intercepts an arc twice its measure - this arc is , which has measure , the corresponding minor arc, will have measure . This gives us , but no clue that yields the area.

Now assume both statements are true. The area is  and the shaded sector is  of the circle, so the area can be calculated to be .

### Example Question #5 : Dsq: Calculating The Area Of A Sector

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

Statement 1: The sector with central angle  has area .

Statement 2: .

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. No clues are given about the measure of , so that of , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle  , or, subsequently, , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let  be the radius of the circle and  be the measure of . Then:

and

The statements can be simplified as

and

From these two statements:

; the second statement can be solved for :

.

, so .

Since , the circle has area . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

.

### Example Question #6 : Dsq: Calculating The Area Of A Sector

What is the area of a  sector of a circle?

Statement 1: The diameter of the circle is 48 inches.

Statement 2: The length of the arc is  inches.

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

BOTH statements TOGETHER are insufficient to answer the question.

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

EITHER 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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The area of a  sector of radius  is

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the  arc is

Given that length, you can find the radius:

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

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

How far has the tip of the hour hand of a clock traveled since noon?

1) It is now 5:00 PM.

2) The hour hand is half the length of the minute hand.

BOTH statements TOGETHER are NOT sufficient to answer the question.

EITHER Statement 1 ALONE or Statement 2 ALONE is sufficient to answer the question.

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

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

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

BOTH statements TOGETHER are NOT sufficient to answer the question.

Explanation:

The time alone is insufficient without the length of the hand. The second statement does not give us that information, only the relationship between the lengths of the two hands, which is useless without the length of the minute hand.

The answer is that both statements together are insufficient to answer the question.

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

Arc  is located on circle  ;  is located on . Which arc, if either, has greater degree measure?

Statement 1:  and  have the same length.

Statement 2:

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

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

EITHER 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:

The arc with the greater degree measure is the one which is the greater part of its circle.

If both arcs have the same length, then the one that is the greater part of its circle must be the one on the smaller circle; Statement 1 alone tells us both have the same length, but not which circle is smaller.

If  , then  has the greater radius and, sqbsequently, the greater circumference; it is the larger circle. But we know nothing about the measures of the arcs, so Statement 2 alone is insufficient.

If we know both statements, however, we know that, since the arcs have the same length, and  is the larger circle - with greater circumference -  must be take up the lesser portion of its circle, and have the lesser degree measure of the two.

### Example Question #51 : Circles

Arc  is located on circle  ;  is located on   Which arc, if either, is longer?

Statement 1:

Statement 2:

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

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.

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:

and  are the central angles that intercept  and , respectively. The measure of an arc is equal to that of its central angle, so, is we are given Statement 2, that , we know that . The arcs are the same portion of their respective circles. The larger of   and  determines which arc is longer; this is given in Statement 1, since, if , then  has the greater radius and circumference.

Both statements together are sufficient to show that  is the longer of the two, but neither alone is suffcient. From Statement 1, the relative sizes of the circles are known, but not the degree measures of the arcs; it is possible for an arc on a larger circle to have length less than, equal to, or greater than the arc on the smaller circle. From Statement 2 alone, the degree measures of the arcs can be proved equal, but not thier lengths.

### Example Question #4 : Dsq: Calculating The Length Of An Arc

Note: Figure NOT drawn to scale.

In the above figure,  is the center of the circle. Give the length of arc .

Statement 1:  is an equilateral triangle.

Statement 2:  has area .

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.

Explanation:

From Statement 1 alone, , so  can be determined to be a  arc. But no method is given to find the length of the arc.

From Statement 2 alone, neither  nor radius  can be determined, as the area of a triangle alone cannot be used to determine any angle or side.

From the two statements together, , and the common sidelength of the equilateral triangle can be determined from the formula

This sidelength  is the radius of the circle. Once  is calculated, the circumference can be calculated, and the arc length will be  of this.

### Example Question #5 : Dsq: Calculating The Length Of An Arc

Note: Figure NOT drawn to scale.

In the above figure,  is the center of the circle. Give the length of arc .

Statement 1:

Statement 2:

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

BOTH statements TOGETHER are insufficient 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 insufficient to answer the question.

Explanation:

If either or both Statement 1 and Statement 2 are known, then then only thing about  that can be determined is that it is an arc of measure . Without knowing any of the linear measures of the circle, such as the radius or the circumference, it is impossible to determine the length of .

### Example Question #51 : Circles

Note: Figure NOT drawn to scale.

In the above figure,  is the center of the circle. Give the length of arc .

Statement 1:

Statement 2: Major arc  has length .

BOTH statements TOGETHER are insufficient to answer the question.

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

EITHER statement ALONE is sufficient to answer the question.

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