ISEE Upper Level Quantitative : Circles
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Example Questions
Example Question #1 : How To Find Circumference
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.
(b) is greater
(a) and (b) are equal
(a) is greater
It is impossible to tell from the information given
(a) is 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.
Example Question #1 : How To Find Circumference
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
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 
Example Question #4 : How To Find Circumference
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
It is impossible to tell from the information given
(b) is greater
(a) and (b) are equal
(a) is greater
(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.
Example Question #26 : Radius
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
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 
Example Question #27 : Radius
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)
Between Points C and D
Between Points E and A
Between Points D and E
Between Points A and B
Between Points B and C
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.
Example Question #161 : Plane Geometry
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
(b) is the greater quantity
(a) is the greater quantity
(a) and (b) are equal
It is impossible to determine which is greater from the information given
(a) and (b) are 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.
Example Question #1 : How To Find The Angle For A Percentage Of A 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 
Example Question #2 : How To Find The Angle For A Percentage Of A 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 
Example Question #3 : How To Find The Angle For A Percentage Of A 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
Example Question #1 : Sectors
Circle A has twice the radius of Circle B. Which is the greater quantity?
(a) The area of a 
(b) The area of Circle B
(b) is greater.
It is impossible to tell from the information given.
(a) and (b) are equal.
(a) is greater.
(a) and (b) are equal.
Let 
A 
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