Circles - GMAT Quantitative Reasoning
Card 0 of 20
Square S is inscribed in circle C as in the figure above. What is the circumference of C?
(1) The perimeter of S is 16.
(2) The area of S is 36.
Square S is inscribed in circle C as in the figure above. What is the circumference of C?
(1) The perimeter of S is 16.
(2) The area of S is 36.
From statement (1), we know that the side of S is 4, and then we can calculate the diagonal of S using the Pythagorean theorem: 
. The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using 
. From statement (2), we know that the side of S is 6, and then we can calculate the diagonal of S using the Pythagorean theorem:
The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using 
.
From statement (1), we know that the side of S is 4, and then we can calculate the diagonal of S using the Pythagorean theorem:
The diagonal of S is the diameter of C. Therefore, we can calculate the circumference by using
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What is the circumference of circle J?
I) Circle J has an area of 
.
II) Circle J has a diameter of 
.
What is the circumference of circle J?
I) Circle J has an area of
II) Circle J has a diameter of
We are given the area and diameter of a circle and asked to find the circumference. We know that diameter is twice the length of a radius, so we also have our radius.
Given the following equations:
We can see that knowing either diameter or area will allow us to find the circumference.
Thus: Each statement alone is enough to solve the question.
We are given the area and diameter of a circle and asked to find the circumference. We know that diameter is twice the length of a radius, so we also have our radius.
Given the following equations:
We can see that knowing either diameter or area will allow us to find the circumference.
Thus: Each statement alone is enough to solve the question.
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What is the circumference of the circle given by:
I) 
.
II) The slope of the tangent to the circle at 
is undefined.
What is the circumference of the circle given by:
I)
II) The slope of the tangent to the circle at
All we need to find circumference is the radius.
I) Gives us the radius squared, so we could find circumference with I.
II) Tells us the slope of the tangent line at a given point is undefined. Only vertical lines have undefined slope. The tangent line is perpendicular to the radius, so we can find our radius by drawing a picture and comparing the location of the center to the location of the tangent line.
So either statement will be sufficient.
All we need to find circumference is the radius.
I) Gives us the radius squared, so we could find circumference with I.
II) Tells us the slope of the tangent line at a given point is undefined. Only vertical lines have undefined slope. The tangent line is perpendicular to the radius, so we can find our radius by drawing a picture and comparing the location of the center to the location of the tangent line.
So either statement will be sufficient.
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What is the circumference of Circle 
?
1.) The diameter of the circle is 
.
2.) The area of the circle is 
.
What is the circumference of Circle
1.) The diameter of the circle is
2.) The area of the circle is
We are asked to find the circumference 
of Circle 
and are given the diameter and the area. We also know that 
. Taking each statement individually:
1.) The diameter 
is 
and we know that the radius 
, so 
. Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.
2.) The area 
of Circle 
is 
, so we can determine that the radius 
. Since the circumference 
, Statement 2 is is sufficient to solve for the circumference of the circle by itself.
We are asked to find the circumference
1.) The diameter
2.) The area
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What is the circumference of Circle 
?
1.) The radius of the circle is 
.
2.) The circle is inside another circle of area 
.
What is the circumference of Circle
1.) The radius of the circle is
2.) The circle is inside another circle of area
We are asked to find the circumference 
of Circle 
and are given the diameter and the area. We also know that 
. Taking each statement individually:
1.) The radius 
is 
and we know that 
. Therefore, Statement 1 is sufficient to solve for the circumference of the circle by itself.
2.) The area 
of the outside circle is 
, but we cannot use this to determine the circumference of Circle 
because we don't know where 
is inside the larger outside circle.
We are asked to find the circumference
1.) The radius
2.) The area
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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 
of the total cake.
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
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.
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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One slice of a pizza is 
. What is the central angle of one slice?
I) Each slice is 
of the whole pizza.
II) Each straight edge the slice is 
inches.
One slice of a pizza is
I) Each slice is
II) Each straight edge the slice is
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.
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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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?
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?
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:
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:
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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.
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.
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 
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:
Thus, Statement II is sufficient, but Statement I is not.
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
Using Statement II if the slice is one-fifth of the total pizza, then we can do the following to find the answer:
Thus, Statement II is sufficient, but Statement I is not.
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Find the angle for the percentage of a circle.
Statement 1: A circle diameter is 5.
Statement 2: The sector length is 
.
Find the angle for the percentage of a circle.
Statement 1: A circle diameter is 5.
Statement 2: The sector length is
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 
.
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 
. We also cannot assume that the sector is a full circle to make any further conclusions.
Therefore, the answer is:
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
Statement 2): The sector length is
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
Therefore, the answer is:
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What time is it?
Statement 1: The minute hand and the hour hand are currently forming a 
angle.
Statement 2: The minute hand is on the 6.
What time is it?
Statement 1: The minute hand and the hour hand are currently forming a
Statement 2: The minute hand is on the 6.
Since there are twelve numbers on the clock, the angular measure from one number to the next is 
; this means 
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.
Since there are twelve numbers on the clock, the angular measure from one number to the next is
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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What time is it?
Statement 1: The minute hand and the hour hand form a 
angle.
Statement 2: The minute hand is exactly on the 8.
What time is it?
Statement 1: The minute hand and the hour hand form a
Statement 2: The minute hand is exactly on the 8.
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 
from one number to the next, so the 8:00 position is the 
position. If the hour hand makes a 
with the minute hand, then the hour hand is either at 
or 
. 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 
, then it is at the 
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 
, then it is at the 
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.
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
Case 1: If the hour hand is at
Case 2: If the hour hand is at
Therefore, only the first case is possible, and if we are given both statements, we know it is 5:40.
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What time is it?
Statement 1: The minute hand and the hour hand are currently forming a 
angle.
Statement 2: The tip of the minute hand has traveled exactly eight inches since last leaving the 12 position.
What time is it?
Statement 1: The minute hand and the hour hand are currently forming a
Statement 2: The tip of the minute hand has traveled exactly eight inches since last leaving the 12 position.
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.
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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Note: Figure NOT drawn to scale.
Refer to the above figure. What is the degree measure of 
?
Statement 1: 
.
Statement 2: 
measures 
.
Note: Figure NOT drawn to scale.
Refer to the above figure. What is the degree measure of
Statement 1:
Statement 2:
From Statement 1, the measure of the arc can be determined by doubling the measure of the intercepting angle, which is 
. From Statement 2, the measure of the arc can be calculated by subtracting from 
the degree measure of the corresponding major arc, which is 
.
From Statement 1, the measure of the arc can be determined by doubling the measure of the intercepting angle, which is
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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)?
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)?
The number of people who voted:
245 people voted for Starr, so the sector representing Starr will have measure
The number of people who voted:
245 people voted for Starr, so the sector representing Starr will have measure
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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?
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?
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 
; that of the square is 
. Therefore, a sector of the circle with area 
will be 
of the circle, which is a sector of measure
The radius of Circle A and the length of a side of the square are the same - we will call each
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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)?
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)?
The number of people who voted:
176 people voted for Starr, so the sector representing Starr will have measure
The number of people who voted:
176 people voted for Starr, so the sector representing Starr will have measure
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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)?
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)?
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
Based on the new information,
people voted.
213 voted for Douglas, meaning that his sector will have degree measure
The reduction in degree measure will be 
.
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
Based on the new information,
213 voted for Douglas, meaning that his sector will have degree measure
The reduction in degree measure will be
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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)?
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)?
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
votes total,
181 of which went to Lealand, so his sector will have measure
,
an increase of 
.
The correct response is 
.
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
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Circle A has radius six times that of Circle B; a sector of Circle A with angle measure 
has the same area as Circle B. Evaluate 
.
Circle A has radius six times that of Circle B; a sector of Circle A with angle measure
Let 
be the radius of Circle B. Then Circle A has radius 
and, subsequently, area 
. Since the area of Circle B is 
, 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
Let
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
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