Circles - GMAT Quantitative
Card 0 of 712
Note: figure NOT drawn to scale.
Give the radius of the above circle with center 
.
Statement 1: The shaded sector has area 
.
Statement 2: Arc 
has length 
.
Note: figure NOT drawn to scale.
Give the radius of the above circle with center
Statement 1: The shaded sector has area
Statement 2: Arc
Let 
be the measure of 
and 
be radius.
From Statement 1 alone, the area of the shaded sector is
However, we have no other information, so we cannot determine the value of the radius.
From Statement 1 alone, the length of the arc of the shaded sector is
Again, we have no other information, so we cannot determine the value of the radius.
Assume both statements hold. From Statements 1 and 2, we have, respectively,
and
If we divide, we get the radius:
.
Let
From Statement 1 alone, the area of the shaded sector is
However, we have no other information, so we cannot determine the value of the radius.
From Statement 1 alone, the length of the arc of the shaded sector
Again, we have no other information, so we cannot determine the value of the radius.
Assume both statements hold. From Statements 1 and 2, we have, respectively,
and
If we divide, we get the radius:
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Right triangle 
is inscribed inside a circle. What is the radius of the circle?
Statement 1: 
Statement 2: 
Right triangle
Statement 1:
Statement 2:
If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius.
Statement 1 alone does not give the hypotenuse of the triangle, or, for that matter, any of the sidelengths. Statement 2 alone gives one sidelength, but does not state whether it is the hypotenuse or not.
Assume both statements are true. Since 
in right triangle 
, then either 
and 
, or vice versa. In either event, 
, being opposite the 
angle, is the short leg of a 30-60-90 triangle, and, by the 30-60-90 Theorem, the hypotenuse is twice its length. This is twice 18, or 36. This is the diameter of the circle, and the radius is half this, or 18.
If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius.
Statement 1 alone does not give the hypotenuse of the triangle, or, for that matter, any of the sidelengths. Statement 2 alone gives one sidelength, but does not state whether it is the hypotenuse or not.
Assume both statements are true. Since
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Parallelogram 
is inscribed inside a circle. What is the radius of the circle?
Statement 1: Each side of Parallelogram 
has length 20.
Statement 2: 
.
Parallelogram
Statement 1: Each side of Parallelogram
Statement 2:
Opposite angles of a parallelogram are congruent, and if the parallelogram is inscribed, both angles are inscribed as well. Congruent inscribed angles on the same circle intercept congruent arcs; since the two congruent arcs together comprise a circle, each intercepted arc is a semicricle. This makes the angles right angles, and this forces a parallelogram inscribed in a circle to be a rectangle.
Statement 1 alone tells us that this is also a square, and that its sides have length 20. The diagonal of a square, which is also a diameter of the circle that circumscribes it, has length 
times that of a side, or 
; half this, or 
, is the radius of the circle.
Statement 2 alone gives a diagonal of the rectangle, which, again, is enough to determine the radius of the circle.
Opposite angles of a parallelogram are congruent, and if the parallelogram is inscribed, both angles are inscribed as well. Congruent inscribed angles on the same circle intercept congruent arcs; since the two congruent arcs together comprise a circle, each intercepted arc is a semicricle. This makes the angles right angles, and this forces a parallelogram inscribed in a circle to be a rectangle.
Statement 1 alone tells us that this is also a square, and that its sides have length 20. The diagonal of a square, which is also a diameter of the circle that circumscribes it, has length
Statement 2 alone gives a diagonal of the rectangle, which, again, is enough to determine the radius of the circle.
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Rectangle is inscribed inside a circle. What is the radius of the circle?
Statement 1: 
Statement 2: 
Rectangle
Statement 1:
Statement 2:
The diameter of a circle that circumscribes a rectangle is equal to the length of a diagonal of the rectangle; the radius is equal to half this.
The two statements together, however, do not yield this. The opposite sides of a rectangle are congruent, so the two statements are actually equivalent; each gives the same dimension of the rectangle. This is insufficient to determine the length of the diagonal.
The diameter of a circle that circumscribes a rectangle is equal to the length of a diagonal of the rectangle; the radius is equal to half this.
The two statements together, however, do not yield this. The opposite sides of a rectangle are congruent, so the two statements are actually equivalent; each gives the same dimension of the rectangle. This is insufficient to determine the length of the diagonal.
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Give the radius of a circle on the coordinate plane.
Statement 1: Of the three intercepts of the circle, exactly two are 
-intercepts, one of which is at the point 
.
Statement 2: Of the three intercepts of the circle, exactly two are 
-intercepts, one of which is at the point 
.
Give the radius of a circle on the coordinate plane.
Statement 1: Of the three intercepts of the circle, exactly two are
Statement 2: Of the three intercepts of the circle, exactly two are
Statement 1 alone only gives one point through which the circle passes, so no information can be determined about the other points or about the size or location of the circle. A similar argument holds for the insufficiency of Statement 2.
Now assume both statements are true. The circle has exactly three intercepts, but it is given that there are two 
-intercepts - 
and one other point - and two 
-intercepts - 
and one other point. The unidentified 
-intercept and the unidentified 
-intercept must be one and the same, and the only possible way this can happen is for this common point to be the origin 
. Since three points define a circle, we can now identify the unique circle through the points 
, 
, and 
, and we can figure out its radius.
Statement 1 alone only gives one point through which the circle passes, so no information can be determined about the other points or about the size or location of the circle. A similar argument holds for the insufficiency of Statement 2.
Now assume both statements are true. The circle has exactly three intercepts, but it is given that there are two
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The equation of a circle can be written in the form
Give the radius of the circle of this equation.
Statement 1: 
Statement 2: 
The equation of a circle can be written in the form
Give the radius of the circle of this equation.
Statement 1:
Statement 2:
The actual form of the equation of a circle is
,
where 
is the location of the center, and 
is the radius.
The radius of the circle in the equation
is therefore 
, making Statement 2 alone sufficient to answer the question - and Statement 1 unhelpful.
The actual form of the equation of a circle is
where
The radius of the circle in the equation
is therefore
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The equation of a circle can be written in the form
Give the radius of the circle of this equation.
Statement 1: 
Statement 2: 
The equation of a circle can be written in the form
Give the radius of the circle of this equation.
Statement 1:
Statement 2:
The actual form of the equation of a circle is
,
where 
is the location of the center, and 
is the radius.
The radius of the circle in the equation
is therefore 
. We need to know the value of 
in the equation.
Assume both statements are true. Then we can add the equations to get 
:
But without further information, we cannot determine 
.
The actual form of the equation of a circle is
where
The radius of the circle in the equation
is therefore
Assume both statements are true. Then we can add the equations to get
But without further information, we cannot determine
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Give the radius of a circle on the coordinate plane.
Statement 1: The circle has its center at 
.
Statement 2: The circle has its 
-intercepts at 
and 
.
Give the radius of a circle on the coordinate plane.
Statement 1: The circle has its center at
Statement 2: The circle has its
Knowing neither the center alone, as given in Statement 1, nor two points alone, as given in Statement 2, is sufficient to find the radius of the circle.
Assume both statements to be true. Knowing the center from Statement 1 and one point on the circle, as given in Statement 2, is enough to determine the radius - use the distance formula to find the distance between the two points and, equivalently, the radius.
Knowing neither the center alone, as given in Statement 1, nor two points alone, as given in Statement 2, is sufficient to find the radius of the circle.
Assume both statements to be true. Knowing the center from Statement 1 and one point on the circle, as given in Statement 2, is enough to determine the radius - use the distance formula to find the distance between the two points and, equivalently, the radius.
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Give the radius of a circle on the coordinate plane.
Statement 1: A right triangle with a hypotenuse with endpoints 
and 
can be inscribed in the circle.
Statement 2: A right triangle with a leg with endpoints 
and 
can be inscribed in the circle.
Give the radius of a circle on the coordinate plane.
Statement 1: A right triangle with a hypotenuse with endpoints
Statement 2: A right triangle with a leg with endpoints
If a right triangle can be inscribed inside a given circle, then its hypotenuse has a length equal to the diameter of the circle, and the radius of the circle can be calculated as half this. Statement 1 gives sufficient information to find this, since the length of the hypotenuse is the distance between its endpoints 
and 
, which is 
; the diameter of the circle is 20, and the radius is half this, or 10. From Statement 2, we can only find the length of one leg of an inscribed right triangle, so the length of the hypotenuse is still open to question.
If a right triangle can be inscribed inside a given circle, then its hypotenuse has a length equal to the diameter of the circle, and the radius of the circle can be calculated as half this. Statement 1 gives sufficient information to find this, since the length of the hypotenuse is the distance between its endpoints
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Give the radius of a circle on the coordinate plane.
Statement 1: A square whose vertices include 
and 
can be inscribed inside the circle.
Statement 2: A right triangle whose vertices include 
and 
can be inscribed inside the circle.
Give the radius of a circle on the coordinate plane.
Statement 1: A square whose vertices include
Statement 2: A right triangle whose vertices include
Assume Statement 1 alone. The length of a segment with the given endpoints can be calculated using the distance formula. However, it is not clear whether the points are opposite vertices, in which case the segment is a diagonal of the square, or the points are consecutive vertices, in which case the segment is a side of the square, making the diagonal of the square 
times this length. The length of the diagonal of the inscribed square cannot be determined for certain; since the diameter of the circle is equal to the length of the diagonal, the diameter cannot be determined, and since the radius is half this, the radius cannot be determined.
Assume Statement 2 alone. The length of a segment with the endpoints can be calculated using the distance formula. However, it is not clear whether the segment is a hypotenuse of the triangle or not; the diameter of a circle is equal to the length of the hypotenuse of an inscribed right triangle, so knowing this is necessary.
Assume both statements to be true. The two statements together give four points of the circle; since three points uniquely define a circle, the circle can be located; subsequently, the radius can be found.
Assume Statement 1 alone. The length of a segment with the given endpoints can be calculated using the distance formula. However, it is not clear whether the points are opposite vertices, in which case the segment is a diagonal of the square, or the points are consecutive vertices, in which case the segment is a side of the square, making the diagonal of the square
Assume Statement 2 alone. The length of a segment with the endpoints can be calculated using the distance formula. However, it is not clear whether the segment is a hypotenuse of the triangle or not; the diameter of a circle is equal to the length of the hypotenuse of an inscribed right triangle, so knowing this is necessary.
Assume both statements to be true. The two statements together give four points of the circle; since three points uniquely define a circle, the circle can be located; subsequently, the radius can be found.
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Give the radius of the above circle with center .
Statement 1: 
is a 
angle.
Statement 2: 
is a 
angle.
Give the radius of the above circle with center
Statement 1:
Statement 2:
The two statements together only give information about angle measures; arc degree measures can be deduced from this information but not any arc lengths or side lengths. Without this information, we cannot obtain the radius of this circle.
The two statements together only give information about angle measures; arc degree measures can be deduced from this information but not any arc lengths or side lengths. Without this information, we cannot obtain the radius of this circle.
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Between Circle 1 and Circle 2, which has the greater radius?
Statement 1: Circle 1 has as a diameter 
; Circle 2 has as a diameter 
; 
is a right triangle.
Statement 2: 
and 
are the legs of 
.
Between Circle 1 and Circle 2, which has the greater radius?
Statement 1: Circle 1 has as a diameter
Statement 2:
Assume both statements are true. Since we have no way to determine which, if either, of the legs of 
is the longer, we have no way to compare the diameters, and, consequently, no way to compare the radii, of the circles with those segments as diameters.
Assume both statements are true. Since we have no way to determine which, if either, of the legs of
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Right triangle 
is inscribed inside a circle. What is the radius of the circle?
Statement 1: 
Statement 2: 
Right triangle
Statement 1:
Statement 2:
If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius. However, each statement alone gives us the length of only one side, and does not clue us in to which side is the hypotenuse.
Assume both statements are true, however. The hypotenuse must always be the longest side of the right triangle, so if two sides have the same length, they must be the legs. A right triangle with congruent legs is a 45-45-90 triangle, and by the 45-45-90 Theorem, its hypotenuse must measure 
the length of a leg. Since each leg of 
measures 100, the length of the hypotenuse, and the diameter of the circle, are 
, and the radius is half this, or 
.
If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius. However, each statement alone gives us the length of only one side, and does not clue us in to which side is the hypotenuse.
Assume both statements are true, however. The hypotenuse must always be the longest side of the right triangle, so if two sides have the same length, they must be the legs. A right triangle with congruent legs is a 45-45-90 triangle, and by the 45-45-90 Theorem, its hypotenuse must measure
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Between Circle 1 and Circle 2, which has the greater radius?
Statement 1: A right triangle, both of whose legs measures 10, can be inscribed inside Circle 1; a square of perimeter 48 can be inscribed inside Circle 2.
Statement 2: A square of area 100 can be inscribed inside Circle 1; a 30-60-90 triangle, one of whose legs measures 
, can be inscribed in side Circle 2.
Between Circle 1 and Circle 2, which has the greater radius?
Statement 1: A right triangle, both of whose legs measures 10, can be inscribed inside Circle 1; a square of perimeter 48 can be inscribed inside Circle 2.
Statement 2: A square of area 100 can be inscribed inside Circle 1; a 30-60-90 triangle, one of whose legs measures
The length of a diagonal of a square inscribed inside a circle is equal to the diameter of the circle, and half this is the radius. If a right triangle is inscribed inside a circle, The hypotenuse of a right triangle inscribed inside a circle is also a diameter, and half the length is the radius. Since each statement alone mentions an inscribed square in one circle to an inscribed right triangle in the other, we need only compare the length of a diagonal of the former to that of the hypotenuse of the latter.
Assume Statement 1 alone. The square inscribed inside Circle 2 has perimeter 48; its sidelength is one fourth this, or 12, and the length of a diagonal is 
times this, or 
. The hypotenuse of the right triangle inscribed in Circle 1, with both legs equal to 10 - which is a 45-45-90 triangle, being isosceles - is, by the 45-45-90 Theorem, times the length of a leg, or 
. The diagonal of the square inscribed inside Circle 2 is longer than the hypotenuse of the triangle inscribed inside Circle 1, so Circle 2 has the greater radius.
Assume Statement 2 alone. A square of area 100 - and, subsequently, of sidelength the square root of this, or 10 - can be inscribed inside Circle 1; its diagonal will have length this, or 
.
The 30-60-90 triangle inscribed inside Circle 2 has a leg of length 
. However, Statement 2 does not make it clear whether the leg is the shorter leg or the longer leg. If it is the shorter leg, then by the 30-60-90 Theorem, the hypotenuse is twice , or 
; if it is the longer leg, then by the 30-60-90 Theorem, the hypotenuse is 
times , or
.
In the first scenario, since 
, Circle 2 has the longer radius. In the second, since 
(this can be seen by noting that 
and 
), Circle 1 has the greater radius.
The length of a diagonal of a square inscribed inside a circle is equal to the diameter of the circle, and half this is the radius. If a right triangle is inscribed inside a circle, The hypotenuse of a right triangle inscribed inside a circle is also a diameter, and half the length is the radius. Since each statement alone mentions an inscribed square in one circle to an inscribed right triangle in the other, we need only compare the length of a diagonal of the former to that of the hypotenuse of the latter.
Assume Statement 1 alone. The square inscribed inside Circle 2 has perimeter 48; its sidelength is one fourth this, or 12, and the length of a diagonal is
Assume Statement 2 alone. A square of area 100 - and, subsequently, of sidelength the square root of this, or 10 - can be inscribed inside Circle 1; its diagonal will have length
The 30-60-90 triangle inscribed inside Circle 2 has a leg of length
In the first scenario, since
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Between Circle 1 and Circle 2, which has the greater radius?
Statement 1: A 
arc of Circle 1 has length equal to one fourth the circumference of Circle 2.
Statement 2: A 
sector of Circle 2 has area equal to four ninths that of Circle 1.
Between Circle 1 and Circle 2, which has the greater radius?
Statement 1: A
Statement 2: A
Assume Statement 1 alone. A 
arc of Circle 1 has 
the circumference of Circle 1. Since this is also 
the circumference of Circle 2, then, if we let 
be the circumferences,
,
and
.
This gives Circle 2 the greater circumference and, subsequently, the greater radius.
Assume Statement 2 alone. A 
sector of Circle 2 has area 
of Circle 2. Since its area is also equal to 
that of Circle 1, then, if 
are the areas of Circle 1 and Circle 2, respectively, then
,
and
This gives Circle 2 the greater area and, subsequently, the greater radius.
Assume Statement 1 alone. A
and
This gives Circle 2 the greater circumference and, subsequently, the greater radius.
Assume Statement 2 alone. A
and
This gives Circle 2 the greater area and, subsequently, the greater radius.
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Between Circle 1 and Circle 2, which has the greater radius?
Statement 1: Circle 1 has as a diameter 
; Circle 2 has as a diameter 
; 
is a right triangle.
Statement 2: 
is the hypotenuse of 
.
Between Circle 1 and Circle 2, which has the greater radius?
Statement 1: Circle 1 has as a diameter
Statement 2:
Statement 2 is unhelpfful as it gives no information about the circles - only a triangle which is not given in that statement to have any connection to the circles. Statement 1 alone is unhelpful in that it only identifies two sides of a triangle as diameters of the circles without giving their lengths.
Now assume both statements to be true. The hypotenuse of a right triangle, which Statement 2 gives as 
, must be the longest side, so 
has greater length than 
. This means that the circle with 
as a diameter, Circle 1, must have a greater diameter than one with 
as a diameter, Circle 2. Since Circle 1 has the greater diameter, it has the greater radius.
Statement 2 is unhelpfful as it gives no information about the circles - only a triangle which is not given in that statement to have any connection to the circles. Statement 1 alone is unhelpful in that it only identifies two sides of a triangle as diameters of the circles without giving their lengths.
Now assume both statements to be true. The hypotenuse of a right triangle, which Statement 2 gives as
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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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