GMAT Math : Sectors

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Example Question #2 : Dsq: Calculating The Area Of A Sector

Sector

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

Statement 1: The circle has circumference $40 \pi$ .

Statement 2: $m \angle ACB = 30 ^{\circ }$

Possible Answers:

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.

Correct answer:

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 $2 \pi$ to yield the radius, and that can be substituted for in the formula $A = \pi r^{2}$ to find the area. However, it provides no clue that might yield $m \angle AOB$ .

Statement 2 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle that intercepts it - has twice this measure, or $60 ^{\circ }$ . Therefore, Statement 2 alone gives the central angle, but does not yield any clues about the area.

Assume both statements are true. The radius is $40 \pi \div 2 \pi = 20$ and the area is $A = \pi \cdot 20^{2 }= 400 \pi$ . The shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, so the area can be calculated to be $\frac{1}{6} \times 400 \pi = \frac{200 \pi}{3}$ .

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

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: $m \angle ACB = 30 ^{\circ }$

Statement 2: $m \angle BOD = 120 ^{\circ }$

Possible Answers:

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice this measure, or $60 ^{\circ }$ .

Statement 2 alone asserts that $m \angle BOD = 120 ^{\circ }$ . By angle addition, $m\angle AOB = 180 ^{\circ } - m \angle BOD = 180 ^{\circ } - 120 ^{\circ } = 60 ^{\circ }$ .

Either statement alone tells us that the shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, and that the white sector is $\frac{5}{6}$ of it; it can be subsequently calculated that the ratio of the areas is $\frac{5}{6} : \frac{1}{6}$ , or 5:1 .

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

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: AD = 100

Statement 2: $m \angle ACB = 30 ^{\circ }$

Possible Answers:

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

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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 $m \angle ACB = 30 ^{\circ }$ . $\angle ACB$ is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice its measure, or $2 \times 30^{\circ }= 60 ^{\circ }$ . From angle addition, this can be subtracted from $180^{\circ }$ to yield the measure of central angle $\angle BOD$ of the shaded sector, which is $120^{\circ }$ . That makes that sector $\frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\frac{2}{3} : \frac{1}{3}$ , or 2:1 .

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

Sector

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

Statement 1: $m \angle BCD = 120^{\circ }$ .

Statement 2: The circle has circumference $28 \pi$ .

Possible Answers:

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 insufficient 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.

Correct answer:

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 $\angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield radius $28 \pi \div 2 \pi = 14$ , and that can be substituted for in the formula $A = \pi r^{2}$ to find the area: $A = \pi \cdot 14^{2} = 196 \pi$ .

However, it provides no clue that might yield $m \angle BOD$ .

From Statement 2 alone, we can find $m \angle BOD$ . $\angle BCD$ , an inscribed angle, intercepts an arc twice its measure - this arc is $\widehat{BAD}$ , which has measure $240^{\circ }$ . $\widehat{B D}$ , the corresponding minor arc, will have measure $360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$ . This gives us $m \angle BOD$ , but no clue that yields the area.

Now assume both statements are true. The area is $196 \pi$ and the shaded sector is $\frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$ .

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

Sector

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

Statement 1: The sector with central angle $\angle BOD$ has area $48 \pi$ .

Statement 2: $\widehat{BD }= 8 \pi$ .

Possible Answers:

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 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.

Correct answer:

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 $\angle BOD$ , so that of $\angle AOB$ , 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 $\widehat{BD }$ , or, subsequently, $\widehat{AB}$ , 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 $N ^{\circ }$ be the measure of $\angle BOD$ . Then:

$\frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and

$\frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$r ^{2}N = 17,280$

and

r N= 1,440

From these two statements:

$\frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

r = 12 ; the second statement can be solved for :

12 N= 1,440

N = 120 .

$m \angle BOD = 120^{\circ }$ , so $m \angle AOB = 60^{\circ }$ .

Since r = 12 , the circle has area $A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$ . 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

$\frac{1}{6} \times 144 \pi = 24\pi$ .

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Example Question #7 : Dsq: Calculating The Area Of A Sector

What is the area of a $60^{\circ}$ sector of a circle?

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

Statement 2: The length of the arc is $8 \pi$ inches.

Possible Answers:

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 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The area of a $60^{\circ}$ sector of radius is

$A= \frac{60}{360} \pi r^{2} = \frac{1}{6} \pi r^{2}$

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 $60^{\circ}$ arc is

$L= \frac{60}{360} \cdot 2 \pi r = \frac{\pi r}{3}$

Given that length, you can find the radius:

$8 \pi = \frac{\pi r}{3}$

24=r

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.

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Example Question #11 : Sectors

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.

Possible Answers:

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.

EITHER Statement 1 ALONE or Statement 2 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 Statement 1 ALONE is not sufficient.

Correct answer:

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.

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Example Question #2 : Dsq: Calculating The Length Of An Arc

Arc $\widehat{AB}$ is located on circle $\odot O$ ; $\widehat{CD}$ is located on $\odot P$ . Which arc, if either, has greater degree measure?

Statement 1: $\widehat{AB}$ and $\widehat{CD}$ have the same length.

Statement 2: OA > PC

Possible Answers:

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.

Correct answer:

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 OA > PC , then $\odot O$ 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 $\odot O$ is the larger circle - with greater circumference - $\widehat{AB}$ must be take up the lesser portion of its circle, and have the lesser degree measure of the two.

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

Arc $\widehat{AB}$ is located on circle $\odot O$ ; $\widehat{CD}$ is located on $\odot P$ Which arc, if either, is longer?

Statement 1: $OA = 2 \cdot PC$

Statement 2: $m \angle AOB = m \angle CPD$

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

$\angle AOB$ and $\angle CPD$ are the central angles that intercept $\widehat{AB}$ and $\widehat{CD}$ , respectively. The measure of an arc is equal to that of its central angle, so, is we are given Statement 2, that $m \angle AOB = m \angle CPD$ , we know that $m \widehat{AB} = m \widehat{CD}$ . The arcs are the same portion of their respective circles. The larger of $\odot O$ and $\odot P$ determines which arc is longer; this is given in Statement 1, since, if $OA = 2 \cdot PC$ , then $\odot O$ has the greater radius and circumference.

Both statements together are sufficient to show that $\widehat{AB}$ 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.

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

Chord

Note: Figure NOT drawn to scale.

In the above figure, is the center of the circle. Give the length of arc $\widehat{AB}$ .

Statement 1: $\Delta OAB$ is an equilateral triangle.

Statement 2: $\Delta OAB$ has area $25 \sqrt{3}$ .

Possible Answers:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

From Statement 1 alone, $m \angle AOB = 60 ^{\circ }$ , so $\widehat{AB}$ can be determined to be a $60 ^{\circ }$ arc. But no method is given to find the length of the arc.

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

From the two statements together, $m \angle AOB = 60 ^{\circ }$ , and the common sidelength of the equilateral triangle can be determined from the formula

$A = \frac{s^{2} \sqrt{3}}{4}$

This sidelength is the radius of the circle. Once is calculated, the circumference can be calculated, and the arc length will be $\frac{60}{360} = \frac{1}{6}$ of this.

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GMAT Math : Sectors

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #2 : Dsq: Calculating The Area Of A Sector

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

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

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

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

Example Question #7 : Dsq: Calculating The Area Of A Sector

Example Question #11 : Sectors

Example Question #2 : Dsq: Calculating The Length Of An Arc

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

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

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