Sectors - GMAT Quantitative
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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.
What is the area of a
Statement 1: The diameter of the circle is 48 inches.
Statement 2: The length of the arc is
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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.
The area of a
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
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.
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 
square centimeters.
Statement 2: The rectangle has perimeter 60 centimeters.
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
Statement 2: The rectangle has perimeter 60 centimeters.
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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 
; the total area of the two black quarter circles is 
, so the area of the white region is their difference,
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 
centimeters, then we can deduce 
using this equation:
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.
The width of the rectangle is equal to the radius of the quarter circles, which we call
The area of the rectangle is
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
If we know that the perimeter of the rectangle is 60 centimeters, we can deduce
Either statement alone allows us to find the radius and, consequently, the area of the white region.
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: 
.
The circle in the above diagram has center
Statement 1: The sector with central angle
Statement 2:
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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
.
Assume Statement 1 alone. No clues are given about the measure of
Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle
Now assume both statements are true. Let
and
The statements can be simplified as
and
From these two statements:
Since
The circle in the above diagram has center 
. Give the area of the shaded sector.
Statement 1: 
.
Statement 2: The circle has circumference 
.
The circle in the above diagram has center
Statement 1:
Statement 2: The circle has circumference
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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 
.
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
Statement 1 alone gives us the circumference; this can be divided by
However, it provides no clue that might yield
From Statement 2 alone, we can find
Now assume both statements are true. The area is
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: 
The circle in the above diagram has center
Statement 1:
Statement 2:
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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 
.
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
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: 
The circle in the above diagram has center
Statement 1:
Statement 2:
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Statement 1 alone asserts that 
. This is an inscribed angle that intercepts the arc 
; therefore, the arc - and the central angle 
that intercepts it - has twice this measure, or 
.
Statement 2 alone asserts that 
. By angle addition, 
.
Either statement alone tells us that the shaded sector is 
of the circle, and that the white sector is 
of it; it can be subsequently calculated that the ratio of the areas is 
, or 
.
Statement 1 alone asserts that
Statement 2 alone asserts that
Either statement alone tells us that the shaded sector is
The circle in the above diagram has center 
. Give the area of the shaded sector.
Statement 1: The circle has circumference 
.
Statement 2: 
The circle in the above diagram has center
Statement 1: The circle has circumference
Statement 2:
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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 the radius, and that can be substituted for 
in the formula 
to find the area. However, it provides no clue that might yield 
.
Statement 2 alone asserts that 
. This is an inscribed angle that intercepts the arc 
; therefore, the arc - and the central angle that intercepts it - has twice this measure, or 
. 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 
and the area is 
. The shaded sector is 
of the circle, so the area can be calculated to be 
.
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
Statement 2 alone asserts that
Assume both statements are true. The radius is
How far has the tip of the hour hand of a clock traveled since noon?
-
It is now 5:00 PM.
-
The hour hand is half the length of the minute hand.
How far has the tip of the hour hand of a clock traveled since noon?
-
It is now 5:00 PM.
-
The hour hand is half the length of the minute hand.
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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.
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.
Arc 
is located on circle 
; 
is located on 
Which arc, if either, is longer?
Statement 1: 
Statement 2: 
Arc
Statement 1:
Statement 2:
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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.
Both statements together are sufficient to show that
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: 
Arc
Statement 1:
Statement 2:
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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.
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
If we know both statements, however, we know that, since the arcs have the same length, and
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: 
Note: Figure NOT drawn to scale.
In the above figure,
Statement 1:
Statement 2:
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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 
.
If either or both Statement 1 and Statement 2 are known, then then only thing about
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 
.
Note: Figure NOT drawn to scale.
In the above figure,
Statement 1:
Statement 2:
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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.
From Statement 1 alone,
From Statement 2 alone, neither
From the two statements together,
This sidelength
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 
.
Note: Figure NOT drawn to scale.
In the above figure,
Statement 1:
Statement 2: Major arc
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Statement 1 only establishes that 
is one-third of the circle. Without other information such as the radius, the circumference, or the length of an arc, it is impossible to determine the length of the chord. Statement 2 alone is also insufficient to give the length of the chord, for similar reasons.
The two statements together, however, establish that 
is the length of the major arc of a 
central angle, and therefore, two-thirds the circumference. The circumference can therefore be calculated to be 
, and minor arc 
is one third of this, or 
.
Statement 1 only establishes that
The two statements together, however, establish that
Note: Figure NOT drawn to scale.
In the above figure, 
is the center of the circle, and 
. Give the length of arc 
.
Statement 1: 
.
Statement 2: The area of 
is 
.
Note: Figure NOT drawn to scale.
In the above figure,
Statement 1:
Statement 2: The area of
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, making 
a right triangle. Since 
, both segments being radii, 
is also a 45-45-90 triangle.
From Statement 1 alone, it can be determined by way of the 45-45-90 Theorem that 
.
From Statement 2 alone, since the area of a right triangle is half the product of its legs,
Since 
and 
,
From either statement alone, the radius of the circle can be calculated. From there, the circumference can be calculated, and the length of the arc can be found by multiplying the circumference by 
From Statement 1 alone, it can be determined by way of the 45-45-90 Theorem that
From Statement 2 alone, since the area of a right triangle is half the product of its legs,
Since
From either statement alone, the radius of the circle can be calculated. From there, the circumference can be calculated, and the length of the arc can be found by multiplying the circumference by
Jane wants to put a sector-shaped sticker on the bottom of her cylindrical water bottle. The water bottle has a circular base. Find the percent of the base that the sticker covers
I) The sticker has a thickness of 
.
II) The central angle of the sticker has a measure of 
.
Jane wants to put a sector-shaped sticker on the bottom of her cylindrical water bottle. The water bottle has a circular base. Find the percent of the base that the sticker covers
I) The sticker has a thickness of
II) The central angle of the sticker has a measure of
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Jane wants to put a sector-shaped sticker on the bottom of her cylindrical water bottle. The water bottle has a circular base. Find the percent of the base that the sticker covers
I) The sticker has a thickness of 
II) The central angle of the sticker has a measure of 
I) is irrelevant. The thickness of the sticker doesn't help us find the area it covers. Statment I) is trying to distract you with 3 dimensions, when we only need to worry about 2.
II) is much more helpful. If we know the central angle, we can find the percentage the sector covers. To find the "answer" perform the following:
So we are dealing with 
of the circle, or 41.67%
Jane wants to put a sector-shaped sticker on the bottom of her cylindrical water bottle. The water bottle has a circular base. Find the percent of the base that the sticker covers
I) The sticker has a thickness of
II) The central angle of the sticker has a measure of
I) is irrelevant. The thickness of the sticker doesn't help us find the area it covers. Statment I) is trying to distract you with 3 dimensions, when we only need to worry about 2.
II) is much more helpful. If we know the central angle, we can find the percentage the sector covers. To find the "answer" perform the following:
So we are dealing with
Dirk bought a sticker to cover part of the base of his scuba tank. The sticker and the base have the same radius. Given the following, find the percentage of the base that the sticker covers.
I) The central angle of the sticker is 
II) The base of the tank has a radius of 5 inches.
Dirk bought a sticker to cover part of the base of his scuba tank. The sticker and the base have the same radius. Given the following, find the percentage of the base that the sticker covers.
I) The central angle of the sticker is
II) The base of the tank has a radius of 5 inches.
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To find the percentage that the sticker covers, we need to know what portion of the circle it covers. We could find this either with the area of the circle or the area of the sector, but we don't have a good way to find the area of the sector.
Statement I gives us the degree measurement of the sticker. We can make a fraction, 
, to find the percentage of the base covered.
Statement II gives us the radius of the base, which lets us also know the radius of the sticker; however, this will not be helpful in finding the percentage of the base covered, so Statement II is not really helpful.
Thus, Statement I is sufficient and Statement II is not.
Recap:
Dirk bought a sticker to cover part of the base of his scuba tank. The sticker and the base have the same radius. Given the following, find the percentage of the base that the sticker covers.
I) The central angle of the sticker is 
II) The base of the tank has a radius of 5 inches
So, according to Statement I, the angle of the sticker is 66 degrees. Because we are dealing with a circle, we can find the percentage of the circle covered by putting 66 over 360.
The sticker covers 
of the scuba tank.
To find the percentage that the sticker covers, we need to know what portion of the circle it covers. We could find this either with the area of the circle or the area of the sector, but we don't have a good way to find the area of the sector.
Statement I gives us the degree measurement of the sticker. We can make a fraction,
Statement II gives us the radius of the base, which lets us also know the radius of the sticker; however, this will not be helpful in finding the percentage of the base covered, so Statement II is not really helpful.
Thus, Statement I is sufficient and Statement II is not.
Recap:
Dirk bought a sticker to cover part of the base of his scuba tank. The sticker and the base have the same radius. Given the following, find the percentage of the base that the sticker covers.
I) The central angle of the sticker is
II) The base of the tank has a radius of 5 inches
So, according to Statement I, the angle of the sticker is 66 degrees. Because we are dealing with a circle, we can find the percentage of the circle covered by putting 66 over 360.
The sticker covers
Consider circle F with sector BFG. What percent of F is BFG?
I) Circle F has a radius of 5 meters and a circumference of 
meters.
II) Sector BFG has a central angle measure of 
.
Consider circle F with sector BFG. What percent of F is BFG?
I) Circle F has a radius of 5 meters and a circumference of
II) Sector BFG has a central angle measure of
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Consider circle F with sector BFG. What percent of F is BFG?
I) Circle F has a radius of 5 meters and a circumference of 
meters
II) Sector BFG has a central angle measure of 
To find the percentage of a circle covered by a sector, we need to know the central angle. A circle will always have 
, so if we put the angle given to us over 
. So, II) allows us to answer the question:
So sector BFG is 87.5% of circle F
I) Give us interesting information, but not helpful in regards to the current question. It doesn't matter what the radius is, and the circumference is also irrelevant.
Therefore, II) is sufficient, but I) is not.
Consider circle F with sector BFG. What percent of F is BFG?
I) Circle F has a radius of 5 meters and a circumference of
II) Sector BFG has a central angle measure of
To find the percentage of a circle covered by a sector, we need to know the central angle. A circle will always have
So sector BFG is 87.5% of circle F
I) Give us interesting information, but not helpful in regards to the current question. It doesn't matter what the radius is, and the circumference is also irrelevant.
Therefore, II) is sufficient, but I) is not.
What is the area of sector AOB?
-
-
Arclength of AB is 
What is the area of sector AOB?
-
-
Arclength of AB is
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The angle 
, so 
.
The Arc length can be related to the radius as follows:
From there, the area of the sector can be found:
The angle
The Arc length can be related to the radius as follows:
From there, the area of the sector can be found:
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.
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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.
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.
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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.