Circles - ISEE Upper Level (grades 9-12) Quantitative Reasoning
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A giant clock has a minute hand five yards in length. Since noon, the tip of the minute hand has traveled 
feet. Which is the greater quantity?
(A) The amount of time that has passed since noon
(B) The amount of time until midnight
A giant clock has a minute hand five yards in length. Since noon, the tip of the minute hand has traveled
(A) The amount of time that has passed since noon
(B) The amount of time until midnight
Five yards is equal to fifteen feet, which is the length of the minute hand. Subsequently, fifteen feet is the radius of the circle traveled by its tip in one hour; the circumference of this circle is 
times this, or
feet.
In one six-hour period, the minute hand revolves six times, so its tip travles six times the circumference, or
The clock has traveled farther than this, so the time is later than 6:00 PM, and more time has elapsed since noon than is left until midnight. This makes (A) greater.
Five yards is equal to fifteen feet, which is the length of the minute hand. Subsequently, fifteen feet is the radius of the circle traveled by its tip in one hour; the circumference of this circle is
In one six-hour period, the minute hand revolves six times, so its tip travles six times the circumference, or
The clock has traveled farther than this, so the time is later than 6:00 PM, and more time has elapsed since noon than is left until midnight. This makes (A) greater.
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Compare the two quantities:
Quantity A: The area of a circle with radius 
Quantity B: The circumference of a circle with radius 
Compare the two quantities:
Quantity A: The area of a circle with radius
Quantity B: The circumference of a circle with radius
Recall for this question that the formulae for the area and circumference of a circle are, respectively:
For our two quantities, we have:
Quantity A: 
Quantity B: 
Therefore, quantity A is greater.
Recall for this question that the formulae for the area and circumference of a circle are, respectively:
For our two quantities, we have:
Quantity A:
Quantity B:
Therefore, quantity A is greater.
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The circumferences of eight circles form an arithmetic sequence. The smallest circle has radius two inches; the second smallest circle has radius five inches. Give the radius of the largest circle.
The circumferences of eight circles form an arithmetic sequence. The smallest circle has radius two inches; the second smallest circle has radius five inches. Give the radius of the largest circle.
The circumference of a circle can be determined by multiplying its radius by 
, so the circumferences of the two smallest circles are
and
The circumferences form an arithmetic sequence with common difference
The circumference of a circle can therefore be found using the formula
where 
and 
; we are looking for that of the 
th smallest circle, so
Since the radius of a circle is the circumference of the circle divided by 
, the radius of this eighth circle is
inches, or 1 foot 11 inches.
The circumference of a circle can be determined by multiplying its radius by
and
The circumferences form an arithmetic sequence with common difference
The circumference of a circle can therefore be found using the formula
where
Since the radius of a circle is the circumference of the circle divided by
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The track at Truman High School is shown above; it is comprised of a square and a semicircle.
Veronica begins at Point A, runs three times around the track counterclockwise, and continues until she reaches Point B. Which of the following comes closest to the distance Veronica runs?
The track at Truman High School is shown above; it is comprised of a square and a semicircle.
Veronica begins at Point A, runs three times around the track counterclockwise, and continues until she reaches Point B. Which of the following comes closest to the distance Veronica runs?
First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 500 feet; this length is about
feet.
The distance around the track is about
feet.
Veronica runs around the track three complete times, for a distance of about
feet.
She then runs from Point A to Point E, which is another 500 feet; Point E to Point D, which is yet another 500 feet, and, finally Point D to Point B, for a final 785 feet. The total distance Veronica runs is about
feet.
Divide by 5,280 to convert to miles:
The closest answer is 
miles.
First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 500 feet; this length is about
The distance around the track is about
Veronica runs around the track three complete times, for a distance of about
She then runs from Point A to Point E, which is another 500 feet; Point E to Point D, which is yet another 500 feet, and, finally Point D to Point B, for a final 785 feet. The total distance Veronica runs is about
Divide by 5,280 to convert to miles:
The closest answer is
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The track at Simon Bolivar High School is a perfect circle of radius 500 feet, and is shown in the above figure. Manuel starts at point C, runs around the track counterclockwise three times, and continues to run clockwise until he makes it to point D. Which of the following comes closest to the number of miles Manuel has run?
The track at Simon Bolivar High School is a perfect circle of radius 500 feet, and is shown in the above figure. Manuel starts at point C, runs around the track counterclockwise three times, and continues to run clockwise until he makes it to point D. Which of the following comes closest to the number of miles Manuel has run?
The circumference of a circle with radius 500 feet is
feet.
Manuel runs this distance three times, then he runs from Point C to D, which is about four-fifths of this distance. Therefore, Manuel's run will be about
feet.
Divide by 5,280 to convert to miles:
,
making 
miles the response closest to the actual running distance.
The circumference of a circle with radius 500 feet is
Manuel runs this distance three times, then he runs from Point C to D, which is about four-fifths of this distance. Therefore, Manuel's run will be about
Divide by 5,280 to convert to miles:
making
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The track at James Buchanan High School is shown above; it is comprised of a square and a semicircle.
Diane wants to run two miles. If she begins at Point A and begins running counterclockwise, when she is finished, which of the five points will she be closest to?
The track at James Buchanan High School is shown above; it is comprised of a square and a semicircle.
Diane wants to run two miles. If she begins at Point A and begins running counterclockwise, when she is finished, which of the five points will she be closest to?
First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is
feet.
The distance around the track is about
feet.
Diane wants to run two miles, or
feet.
She will make about
circuits around the track.
Equivalently, she will run the track 5 complete times for a total of about
feet,
so she will have
feet to go.
She is running counterclockwise, so she will proceed from Point A to Point D, running another 800 feet, leaving
feet.
She will almost, but not quite, finish the 628 feet from Point D to Point B.
The correct response is Point B.
First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is
The distance around the track is about
Diane wants to run two miles, or
She will make about
circuits around the track.
Equivalently, she will run the track 5 complete times for a total of about
so she will have
She is running counterclockwise, so she will proceed from Point A to Point D, running another 800 feet, leaving
She will almost, but not quite, finish the 628 feet from Point D to Point B.
The correct response is Point B.
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The track at Monroe High School is a perfect circle of radius 600 feet, and is shown in the above figure. Quinnella wants to run around the track for one and a half miles. If Quinnella starts at point C and runs counterclockwise, which of the following is closest to the point at which she will stop running?
(Assume the five points are evenly spaced)
The track at Monroe High School is a perfect circle of radius 600 feet, and is shown in the above figure. Quinnella wants to run around the track for one and a half miles. If Quinnella starts at point C and runs counterclockwise, which of the following is closest to the point at which she will stop running?
(Assume the five points are evenly spaced)
A circle of radius 600 feet will have a circumference of
feet.
Quinnella will run one and a half miles, or
feet,
which is about 
times the circumference of the circle.
Quinnella will run around the track twice, returning to Point C; she will not quite make it to Point B a third time, since that is one-fifth of the track, or 0.2. The correct response is that she will be between Points B and C.
A circle of radius 600 feet will have a circumference of
Quinnella will run one and a half miles, or
which is about
Quinnella will run around the track twice, returning to Point C; she will not quite make it to Point B a third time, since that is one-fifth of the track, or 0.2. The correct response is that she will be between Points B and C.
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The track at Monroe Elementary School is a perfect circle of radius 400 feet, and is shown in the above figure.
Evan and his younger brother Mike both start running from Point A. Evan runs counterclockwise, running once around the track and then on to Point E; Mike runs clockwise, meeting Evan at Point E and stopping.
Which of the following is the greater quantity?
(a) Twice Mike's average speed.
(b) Evan's average speed.
(Assume the five points are evenly spaced)
The track at Monroe Elementary School is a perfect circle of radius 400 feet, and is shown in the above figure.
Evan and his younger brother Mike both start running from Point A. Evan runs counterclockwise, running once around the track and then on to Point E; Mike runs clockwise, meeting Evan at Point E and stopping.
Which of the following is the greater quantity?
(a) Twice Mike's average speed.
(b) Evan's average speed.
(Assume the five points are evenly spaced)
It is not actually necessary to know the radius or length of the track if we know the points are equally spaced. Evan runs once around the track counterclockwise and then on to Point E, which is the next point after A; this means he runs around the track 
times. Mike runs around the track clockwise from Point A to Point E, in the same time, meaning he runs around the track 
times.
Therefore, Evan's speed is 
times Mike's speed. As a result, Twice Mike's speed would be greater than Evan's speed, making (b) the greater.
It is not actually necessary to know the radius or length of the track if we know the points are equally spaced. Evan runs once around the track counterclockwise and then on to Point E, which is the next point after A; this means he runs around the track
Therefore, Evan's speed is
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The track at Monroe High School is a perfect circle of radius 600 feet, and is shown in the above figure.
Jerry begins his one-mile run at Point A, then runs counterclockwise around the track. At the end of his one-mile run, which is the greater quantity?
(a) The additional distance he would have to run if he were to continue to run counterclockwise to Point A.
(b) The additional distance he would have to run if he were to turn back and run clockwise to Point A.
The track at Monroe High School is a perfect circle of radius 600 feet, and is shown in the above figure.
Jerry begins his one-mile run at Point A, then runs counterclockwise around the track. At the end of his one-mile run, which is the greater quantity?
(a) The additional distance he would have to run if he were to continue to run counterclockwise to Point A.
(b) The additional distance he would have to run if he were to turn back and run clockwise to Point A.
The circumference of a circle with radius 600 feet is
feet.
A one mile run would be
times the length of the track.
Therefore, Jerry's run takes him around the track once, and about 0.4 times the length of the track. Since he is running counterclockwise, but has not made it halfway around the track yet, the longer of the two paths is to proceed counterclockwise and run the remaining 0.6 of the track. This makes (a) greater.
The circumference of a circle with radius 600 feet is
A one mile run would be
times the length of the track.
Therefore, Jerry's run takes him around the track once, and about 0.4 times the length of the track. Since he is running counterclockwise, but has not made it halfway around the track yet, the longer of the two paths is to proceed counterclockwise and run the remaining 0.6 of the track. This makes (a) greater.
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The radii of six circles form an arithmetic sequence. The radius of the second-smallest circle is twice that of the smallest circle. Which of the following, if either, is the greater quantity?
(a) The circumference of the largest circle
(b) Twice the circumference of the third-smallest circle
The radii of six circles form an arithmetic sequence. The radius of the second-smallest circle is twice that of the smallest circle. Which of the following, if either, is the greater quantity?
(a) The circumference of the largest circle
(b) Twice the circumference of the third-smallest circle
Call the radius of the smallest circle 
. The radius of the second-smallest circle is then 
, and the common difference of the radii is 
.
The radii of the six circles are, from least to greatest:
The largest circle has circumference
The third-smallest circle has circumference:
Twice this is
The circumference of the sixth circle is equal to twice that of the third-smallest circle, so the correct choice is that that (a) and (b) are equal.
Call the radius of the smallest circle
The radii of the six circles are, from least to greatest:
The largest circle has circumference
The third-smallest circle has circumference:
Twice this is
The circumference of the sixth circle is equal to twice that of the third-smallest circle, so the correct choice is that that (a) and (b) are equal.
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In the above figure, 
.
Which is the greater quantity?
(a) The sum of the circumferences of the inner and outer circles
(b) The sum of the circumferences of the second-largest and third-largest circles
In the above figure,
Which is the greater quantity?
(a) The sum of the circumferences of the inner and outer circles
(b) The sum of the circumferences of the second-largest and third-largest circles
For the sake of simplicity, we will assume that 
; this reasoning is independent of the actual length.
The four concentric circles have radii 1, 2, 3, and 4, respectively, and their circumferences can be found by multiplying these radii by 
.
The inner and outer circles have circumferences 
and 
, respectively; the sum of these circumferences is 
. The other two circles have circumferences 
and 
; the sum of these circumferences is 
.
The two sums are therefore equal.
For the sake of simplicity, we will assume that
The four concentric circles have radii 1, 2, 3, and 4, respectively, and their circumferences can be found by multiplying these radii by
The inner and outer circles have circumferences
The two sums are therefore equal.
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What is the angle measure of 
in the figure above if the sector comprises 37% of the circle?
What is the angle measure of
It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle. To do this, you merely need to multiply 
by 
˚. This yields 
˚.
It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle. To do this, you merely need to multiply
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What is the angle measure of 
in the figure above if the sector comprises 
% of the circle?
What is the angle measure of
It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle. To do this, you merely need to multiply 
by 
˚. This yields 
˚.
It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle. To do this, you merely need to multiply
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What is the angle measure of in the figure if the sector comprises 
of the circle?
What is the angle measure of
It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle. To do this, you merely need to multiply by ˚. This yields ˚
It is very easy to compute the angle of a sector if we know what it is as a percentage of the total circle. To do this, you merely need to multiply
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Which is the greater quantity?
(a) The degree measure of a 10-inch-long arc on a circle with radius 8 inches.
(b) The degree measure of a 12-inch-long arc on a circle with radius 10 inches.
Which is the greater quantity?
(a) The degree measure of a 10-inch-long arc on a circle with radius 8 inches.
(b) The degree measure of a 12-inch-long arc on a circle with radius 10 inches.
(a) A circle with radius 8 inches has crircumference 
inches. An arc 10 inches long is 
of that circle. 
, the degree measure of this arc.
(b) A circle with radius 10 inches has crircumference 
inches. An arc 12 inches long is 
of that circle. 
, the degree measure of this arc.
(a) is the greater quantity.
(a) A circle with radius 8 inches has crircumference
(b) A circle with radius 10 inches has crircumference
(a) is the greater quantity.
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Note: figure NOT drawn to scale
Refer to the above figure. Which is the greater quantity?
(a) 
(b) 
Note: figure NOT drawn to scale
Refer to the above figure. Which is the greater quantity?
(a)
(b)
Since
,
the triangle is a right triangle with right angle 
.
is an inscribed angle on the circle, so the arc it intercepts is a semicricle. Therefore, 
is also a semicircle, and it measures 
.
Since
the triangle is a right triangle with right angle
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Refer to the above figure. Which is the greater quantity?
(a) 
(b) 55
Refer to the above figure. Which is the greater quantity?
(a)
(b) 55
The measure of an inscribed angle of a circle is one-half that of the arc it intercepts. Therefore, 
.
The measure of an inscribed angle of a circle is one-half that of the arc it intercepts. Therefore,
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Note: Figure NOT drawn to scale
Refer to the above figure. Which is the greater quantity?
(a) 
(b) 90
Note: Figure NOT drawn to scale
Refer to the above figure. Which is the greater quantity?
(a)
(b) 90
The measure of an arc intercepted by an inscribed angle of a circle is twice that of the angle. Therefore, 
The measure of an arc intercepted by an inscribed angle of a circle is twice that of the angle. Therefore,
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has twice the radius of 
. Sector 1 is part of 
; Sector 2 is part of 
; the two sectors are equal in area.
Which is the greater quantity?
(a) Twice the degree measure of the central angle of Sector 1
(b) The degree measure of the central angle of Sector 2
Which is the greater quantity?
(a) Twice the degree measure of the central angle of Sector 1
(b) The degree measure of the central angle of Sector 2
has twice the radius of 
, so 
has four times the area of 
. This means that for a sector of 
to have the same area as a sector of 
, the central angle of the latter sector must be four times that of the former sector. This makes (b) greater than (a), which is only twice that of the former sector.
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The area of the shaded sector in circle O is 
. What is the angle measure 
?
The area of the shaded sector in circle O is
Remember that the angle for a sector or arc is found as a percentage of the total 
degrees of the circle. The proportion of 
to 
is the same as 
to the total area of the circle.
The area of a circle is found by:
For our data, this means:
Now we can solve for 
using the proportions:
Cross multiply:
Divide both sides by 
:
Therefore, 
is 
˚.
Remember that the angle for a sector or arc is found as a percentage of the total
The area of a circle is found by:
For our data, this means:
Now we can solve for
Cross multiply:
Divide both sides by
Therefore,
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