Sectors - GMAT Quantitative Reasoning

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Question

Brendan's girlfriend made him a cheescake for his birthday. He eats one slice a day. What is the measure of the central angle of each slice?

I) The diameter of the cake is .

II) Each slice is $\frac{1}{12}$ of the total cake.

Answer

In this case we are given a circle and asked to find the angle of a portion of it.

The diameter would allow us to find many things related to the circle, but not an individual slice.

However, knowing that each slice is 1/12 of the total allows us to multiply 360 by 1/12 and find out that each slice is 30 degrees.

Therefore statement II alone is sufficient in answering the question.

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Question

One slice of a pizza is $\frac{5\pi}{4}in^2$ . What is the central angle of one slice?

I) Each slice is of the whole pizza.

II) Each straight edge the slice is $\frac{\sqrt{5}}{2}$ inches.

Answer

I) Gives us the percentage of one slice of the whole pizza. We can take 15% of 360 to find the central angle.

II) Gives us the radius of the pizza. We can use the radius to find the area of the pizza. With the total area and the area of one slice we can find the percentage of the whole and from there, the angle of one slice.

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Question

Walder is at his cousin's wedding preparing to eat a slice of pie.

I) Walder's slice has a radius of half a meter

II) Walder's slice is 5% of the total pie

What is the central angle of Walder's slice?

Answer

The central angle of a sector, in this case represented by the slice of pie, can be thought of as a percentage of the whole circle. Circles have 360 degrees.

Statement I gives us the radius of the circle. We could find the diameter, area, or circumference with this, but not the central angle of that slice.

Statement II gives us the percentage of the whole circle that the slice represents, 5%. We can use this to find the number of degrees in the central angle of the slice, because it will just be 5% of 360.

Thus, Statement II is sufficient, but Statement I is not.

To recap:

Walder is at his cousin's wedding preparing to eat a slice of pie.

I) Walder's slice has a radius of half a meter

II) Walder's slice is 5% of the total pie

What is the central angle of Walder's slice?

Use Statement II to find the angle. The angle must be 5% of 360:

$0.05\cdot 360^{\circ}=18^{\circ}$

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Question

Alex decided to order a circular pizza. Find the angle that represents the slice of pizza he ate.

I) The pizza had a radius of 14in.

II) The slice Alex ate represented one-fifth of the total pizza.

Answer

To find the angle of a sector (in this case, that represented by the slice of pizza), we need to know with how much of the circle we are dealing.

Statement I gives us the radius of the circle. This is helpful for a lot of other things, but not finding our central angle.

Statement II tells us what portion of the pizza we are concerned with. We can multiply $360^{\circ}$ by one-fifth to get the correct answer.

Using Statement II if the slice is one-fifth of the total pizza, then we can do the following to find the answer:

$360^{\circ}\cdot \frac{1}{5}=72^{\circ}$

Thus, Statement II is sufficient, but Statement I is not.

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Question

Find the angle for the percentage of a circle.

Statement 1: A circle diameter is 5.

Statement 2: The sector length is $2\pi$ .

Answer

The question asks to solve the angle of a percentage of a circle.

Statement 1): A circle diameter is 5.

Statement 1) is sufficient to solve for the angle of the circle because the statement itself provides that the shape is a full circle, 360 degrees, and is of the circular sector.

Statement 2): The sector length is $2\pi$ .

Statement 2) does not have sufficient information to solve for the angle. The precentage of the circle is not provided and we do not know how much of the circle will have a sector length of $2\pi$ . We also cannot assume that the sector is a full circle to make any further conclusions.

Therefore, the answer is:

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

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Question

What time is it?

Statement 1: The minute hand and the hour hand are currently forming a $75^{\circ }$ angle.

Statement 2: The minute hand is on the 6.

Answer

Since there are twelve numbers on the clock, the angular measure from one number to the next is $30^{\circ }$ ; this means $75 ^{\circ }$ represents two and a half number positions.

Suppose we know both statements. Since the minute hand is on the 6, the hour hand is either midway between the 3 and the 4, or midway between the 8 and the 9. Both scenarios are possible, as they correspond to 3:30 and 8:30, respectively, so the question is not answered even if we know both statements.

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Question

What time is it?

Statement 1: The minute hand and the hour hand form a $70 ^{\circ }$ angle.

Statement 2: The minute hand is exactly on the 8.

Answer

The first event happens numerous times over the course of twelve hours, so the first statement is not enough to deduce the time; all the second statement tells you is that it is forty minutes after an hour (12:40, 1:40, etc.)

Suppose we put the two statements together. It is $360 \div 12 = 30 ^{\circ }$ from one number to the next, so the 8:00 position is the $30 \cdot 8 = 240 ^{\circ}$ position. If the hour hand makes a $70 ^{\circ }$ with the minute hand, then the hour hand is either at $240 -70 = 170 ^{\circ }$ or $240 +70 = 310 ^{\circ }$ . Since forty minutes is two-thirds of an hour, however, the hour hand must be two-thirds of the way from one number to the next.

Case 1: If the hour hand is at $170 ^{\circ }$ , then it is at the $\frac{170 }{30} = 5 \frac{2}{3}$ position - in other words, two-thirds of the way from the 5 to the 6. This is consistent with our conditions.

Case 2: If the hour hand is at $310 ^{\circ }$ , then it is at the $\frac{310 }{30} = 10 \frac{1}{3}$ position - in other words, one-third of the way from the 10 to the 11. This is inconsistent with our conditions.

Therefore, only the first case is possible, and if we are given both statements, we know it is 5:40.

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Question

What time is it?

Statement 1: The minute hand and the hour hand are currently forming a $45^{\circ }$ angle.

Statement 2: The tip of the minute hand has traveled exactly eight inches since last leaving the 12 position.

Answer

The two statements together are not enough unless you know the size of the minute hand; without this information, you cannot tell the angular position of the minute hand, so, even if you know the angle the hands are making, you do not know the position of the hour hand either.

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Question

Note: Figure NOT drawn to scale.

Refer to the above figure. What is the degree measure of $\widehat{AB }$ ?

Statement 1: $m \angle C = 56^{\circ }$ .

Statement 2: $\widehat{ACB }$ measures $248 ^{\circ }$ .

Answer

From Statement 1, the measure of the arc can be determined by doubling the measure of the intercepting angle, which is $\angle C$ . From Statement 2, the measure of the arc can be calculated by subtracting from $360 ^{\circ }$ the degree measure of the corresponding major arc, which is $\widehat{ACB }$ .

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Question

After a student body election, Henry is constructing a circle graph to represent the above voter count.

What will be the measure of the central angle of the sector representing Starr (nearest whole number)?

Answer

The number of people who voted:

245 people voted for Starr, so the sector representing Starr will have measure

$\frac{245}{880} \times 360^{\circ } \approx 100^{\circ }$

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Question

The radius of Circle A is equal to the sidelength of Square B. A sector of Circle A has the same area as Square B. Which of the following is the degree measure of this sector?

Answer

The radius of Circle A and the length of a side of the square are the same - we will call each . The area of the circle is $A_{a}= \pi r^{2}$ ; that of the square is $A_{b}= r^{2}$ . Therefore, a sector of the circle with area $r^{2}$ will be $\frac{1}{\pi}$ of the circle, which is a sector of measure

$\frac{1}{\pi} \times 360 ^{\circ } = \left (\frac{360}{\pi} \right )^{\circ }$

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Question

After a student body election, Henry is constructing a circle graph to represent the above voter count.

What will be the measure of the central angle of the sector representing Thomas (nearest whole number)?

Answer

The number of people who voted:

176 people voted for Starr, so the sector representing Starr will have measure

$\frac{176}{880} \times 360^{\circ } \approx 72^{\circ }$

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Question

Above are the results of the election for student body president that Greg is about to publish in the school newspaper. At the last minute, his friend Melissa stops him and reminds him that there was a sixth candidate, Wilson, who got 202 votes.

Greg'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 Douglas decrease (nearest whole degree)?

Answer

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

.

213 voted for Douglas, meaning that his sector will have degree measure

$\frac{213}{880} \times 360^{\circ} \approx 87^{\circ }$

Based on the new information,

people voted.

213 voted for Douglas, meaning that his sector will have degree measure

$\frac{213}{1,082} \times 360^{\circ} \approx 71^{\circ }$

The reduction in degree measure will be $87^{\circ } - 71^{\circ } = 16^{\circ }$ .

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Question

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)?

Answer

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

$\frac{101}{880} \times 360^{\circ} = 41.3^{\circ}$

representing Lealand's share of the vote.

However, the corrected figures are

votes total,

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

$\frac{181}{960} \times 360^{\circ} = 67.9^{\circ}$ ,

an increase of $67.9^{\circ } - 41.3^{\circ }= 26.6^{\circ }$ .

The correct response is $27^{\circ }$ .

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Question

Circle A has radius six times that of Circle B; a sector of Circle A with angle measure $N^{\circ }$ has the same area as Circle B. Evaluate .

Answer

Let be the radius of Circle B. Then Circle A has radius and, subsequently, area $\pi \left ( 6r\right )^{2} = 36 \pi r ^{2}$ . Since the area of Circle B is $\pi r^{2}$ , the area of Circle A is 36 times that of Circle B.

The given sector of Circle A has the same area as Circle B, so the sector is one thirty-sixth of the circle. That makes the angle measure of the sector

$\frac{1}{36} \times 360 ^{\circ } = 10^{\circ }$

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Question

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.

Answer

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:

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

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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Question

The above figure shows two quarter circles inscribed inside a rectangle. What is the total area of the white region?

Statement 1: The area of the black region is $50\pi$ square centimeters.

Statement 2: The rectangle has perimeter 60 centimeters.

Answer

The width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is $r \cdot 2r = 2r^{2}$ ; the total area of the two black quarter circles is $2 \cdot \frac{1}{4} \cdot \pi r^{2} = \frac{1}{2} \pi r^{2}$ , so the area of the white region is their difference,

$2r^{2} - \frac{1}{2} \pi r^{2}$

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is $50 \pi$ centimeters, then we can deduce using this equation:

$\frac{1}{2} \pi r^{2} = 50 \pi$

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

Either statement alone allows us to find the radius and, consequently, the area of the white region.

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Question

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

Answer

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

From these two statements:

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

; the second statement can be solved for :

.

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

Since , 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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Question

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 $28 \pi$ .

Answer

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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Question

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

Answer

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 .

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