Circles - ISEE Upper Level Quantitative Reasoning
Card 0 of 380
In the above figure, 
.
Give the ratio of the area of the outer ring to that of the inner circle.
In the above figure,
Give the ratio of the area of the outer ring to that of the inner circle.
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 areas can be found by substituting each radius for 
in the formula 
.
The areas of the largest circle and the second-largest circle are, respectively,
The difference of their areas, which is the area of the outer ring, is
.
The inner circle has area
.
The ratio of these areas is therefore
, or 7 to 1.
For the sake of simplicity, we will assume that
The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for
The areas of the largest circle and the second-largest circle are, respectively,
The difference of their areas, which is the area of the outer ring, is
The inner circle has area
The ratio of these areas is therefore
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The area of a circle is 
. Give the diameter and radius of the circle.
The area of a circle is
The area of a circle can be calculated as 
where 
is the radius of the circle, and 
is approximately 
.
To find the diameter, multiply the radius by 
:
The area of a circle can be calculated as
To find the diameter, multiply the radius by
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The circumference of a circle is 
. Give the diameter of the circle.
The circumference of a circle is
The circumference can be calculated as 
, where 
is the radius of the circle and 
is the diameter of the circle.
The circumference can be calculated as
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If the value of a radius is 
, what is the value of the diameter if the value of 
?
If the value of a radius is
If the value of a radius is 
, and the value of 
, then the radius will be equal to:
Given that the diameter is twice that of the radius, the diameter will be equal to:
This is equal to:
If the value of a radius is
Given that the diameter is twice that of the radius, the diameter will be equal to:
This is equal to:
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A series of circles has the following radius values:
If the diameter is then calculated for this set, what would be the median diameter?
A series of circles has the following radius values:
If the diameter is then calculated for this set, what would be the median diameter?
The median is the middle number in a set when that set is ordered smalles to largest.
When 
is ordered smallest to largest, we get 
Here, the median would be 
.
Given that a diameter is twice the radius, the diamater would be 
(twice the value of 
).
The median is the middle number in a set when that set is ordered smalles to largest.
When
Here, the median would be
Given that a diameter is twice the radius, the diamater would be
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If the area of a circle is equal to 
, then what is the diameter?
If the area of a circle is equal to
If the area of a circle is equal to 
, then the radius is equal to 
.
This is because the equation for the area of a circle is 
.
Thus, 
.
Then the diameter is 12.
If the area of a circle is equal to
This is because the equation for the area of a circle is
Thus,
Then the diameter is 12.
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The circumference of a circle is 
. Give the diameter of the circle.
The circumference of a circle is
The circumference can be calculated as 
, where 
is the radius of the circle and 
is the diameter of the circle.
The circumference can be calculated as
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You have a circular lens with a circumference of 
, find the diameter of the lens.
You have a circular lens with a circumference of
You have a circular lens with a circumference of 
, find the diameter of the lens.
Begin with the circumference of a circle formula.
Now, we know that
Because our radius is half of our diameter.
So, we can change our original formula to be:
Now, we can see that all we need to do is divide our circumference by pi to get our diameter.
Now plug in our known and solve:
So our answer is 16.67inches
You have a circular lens with a circumference of
Begin with the circumference of a circle formula.
Now, we know that
Because our radius is half of our diameter.
So, we can change our original formula to be:
Now, we can see that all we need to do is divide our circumference by pi to get our diameter.
Now plug in our known and solve:
So our answer is 16.67inches
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You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be 
.
What is the diameter of the crater?
You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be
What is the diameter of the crater?
You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be 
.
What is the diameter of the crater?
To solve this, we need to recall the formula for the area of a circle.
Now, we know A, so we just need to plug in and solve for r!
Begin by dividing out the pi
Then, square root both sides.
Now, recall that diameter is just twice the radius.
You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be
What is the diameter of the crater?
To solve this, we need to recall the formula for the area of a circle.
Now, we know A, so we just need to plug in and solve for r!
Begin by dividing out the pi
Then, square root both sides.
Now, recall that diameter is just twice the radius.
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The circumference of a circle is . Give the diameter of the circle.
The circumference of a circle is
The circumference can be calculated as , where is the radius of the circle and is the diameter of the circle.
The circumference can be calculated as
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A circle has a radius of 
. What is the ratio of the diameter to the circumference?
A circle has a radius of
The information about the radius is unnecessary to the problem. The equation of the circumference is:
Therefore, the circumference is 
times larger than the diameter, and the ratio of the diameter to the circumference is:
The information about the radius is unnecessary to the problem. The equation of the circumference is:
Therefore, the circumference is
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A circle has a radius of . What is the ratio of the diameter to the circumference?
A circle has a radius of
The information about the radius is unnecessary to the problem. The equation of the circumference is:
Therefore, the circumference is times larger than the diameter, and the ratio of the diameter to the circumference is:
The information about the radius is unnecessary to the problem. The equation of the circumference is:
Therefore, the circumference is
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At which of the following times on a clock is the smaller angle created by the minute hand and hour hand NOT less than 
?
At which of the following times on a clock is the smaller angle created by the minute hand and hour hand NOT less than
In this problem, you are working with the shape of a clock. A clock is in the shape of a circle, which has 
total. You are looking for the time for which the smaller angle created by the minute and hour hands is NOT less than 
- in other words, the time must create an angle greater than 
.
First, it's helpful to recognize at what times a right angle (
) exists. This would be 3:00 and 9:00. It's also useful to recognize that times such as 3:30 and 9:30 will be slightly under 
- the hour hand moves just past the 3 and 9 on a clock at these times to create this effect. You can draw a clock out to show these angles.
The three times that do create an angle less than 
, and thus are incorrect, are 2:00, 4:30, and 11:00.
2:00 has the minute hand straight up on the 12 and the hour hand straight at the 2, making it less than the 
angle created by 3:00.
4:30 has the hour hand in between the 4 and 5 and the minute hand on the 6, making it less than the 
angle created by 3:30.
11:00 has the minute hand straight up on the 12 and the hour hand straight at the 11, making it less than the 
angle created by 9:00.
1:30 is the angle that is NOT less than 
. The hour hand is in between the 1 and 2, and the minute hand is straight down on the 6. This angle is larger than a perfect right angle of 
. Therefore, 1:30 is the correct answer.
In this problem, you are working with the shape of a clock. A clock is in the shape of a circle, which has
First, it's helpful to recognize at what times a right angle (
The three times that do create an angle less than
2:00 has the minute hand straight up on the 12 and the hour hand straight at the 2, making it less than the
4:30 has the hour hand in between the 4 and 5 and the minute hand on the 6, making it less than the
11:00 has the minute hand straight up on the 12 and the hour hand straight at the 11, making it less than the
1:30 is the angle that is NOT less than
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How many degrees are in an angle formed by the hour and minute hand when it is 5:15?
How many degrees are in an angle formed by the hour and minute hand when it is 5:15?
When it is 5:15, the hour hand is pointed to 5 and the minute hand is pointed to 3.
Therefore, the angle created will span between 3 and 5, which spans "2 hours" worth of time on the clock. Given that there are a total of 12 hours of time on the clock, this means that the angle formed will be equal to 
of the total degrees in the clock.
Since the clock is a circle, it contains 360 degrees.
of 360 is 60; therefore, 60 is the correct answer.
When it is 5:15, the hour hand is pointed to 5 and the minute hand is pointed to 3.
Therefore, the angle created will span between 3 and 5, which spans "2 hours" worth of time on the clock. Given that there are a total of 12 hours of time on the clock, this means that the angle formed will be equal to
Since the clock is a circle, it contains 360 degrees.
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How many degrees are are in the angle formed by the minute and hour hand when it is 7:05?
How many degrees are are in the angle formed by the minute and hour hand when it is 7:05?
When it is 7:05, the hour hand is pointed to 7 and the minute hand is pointed to 1.
Therefore, the angle created will span between 1 and 7, which spans "6 hours" worth of time on the clock. Given that there are a total of 12 hours of time on the clock, this means that the angle formed will be equal to 
of the total degrees in the clock.
Since the clock is a circle, it contains 360 degrees.
of 360 is 180; therefore, 180 is the correct answer.
When it is 7:05, the hour hand is pointed to 7 and the minute hand is pointed to 1.
Therefore, the angle created will span between 1 and 7, which spans "6 hours" worth of time on the clock. Given that there are a total of 12 hours of time on the clock, this means that the angle formed will be equal to
Since the clock is a circle, it contains 360 degrees.
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The hour hand and the minute hand of a clock form a 30-degree angle. If the minute hand reads 55 minutes, what number may the hour hand be pointed to?
The hour hand and the minute hand of a clock form a 30-degree angle. If the minute hand reads 55 minutes, what number may the hour hand be pointed to?
When the minute hand is pointed to 55 minutes (or the 11th hour), the hour hand must be pointed to either 10 or 12.
This is because the angle created will span between 10 and 11 or 11 and 12, which spans "1 hour" worth of time on the clock. Given that there are a total of 12 hours of time on the clock, this means that the angle formed will be equal to 
of the total degrees in the clock.
Since the clock is a circle, it contains 360 degrees.
of 360 is 30; therefore, the answer choice 10 or 12 proves to be correct.
When the minute hand is pointed to 55 minutes (or the 11th hour), the hour hand must be pointed to either 10 or 12.
This is because the angle created will span between 10 and 11 or 11 and 12, which spans "1 hour" worth of time on the clock. Given that there are a total of 12 hours of time on the clock, this means that the angle formed will be equal to
Since the clock is a circle, it contains 360 degrees.
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The hour hand and the minute hand of a clock form a 60-degree angle. If the minute hand reads 45 minutes, what number may the hour hand be pointed to?
The hour hand and the minute hand of a clock form a 60-degree angle. If the minute hand reads 45 minutes, what number may the hour hand be pointed to?
When the hour hand is pointed to 45 minutes (or the 9th hour), the hour hand must be pointed to either 7 or 11.
This is because the angle created will span between 9 and 11 or 9 and 7, which spans "2 hours" worth of time on the clock. Given that there are a total of 12 hours of time on the clock, this means that the angle formed will be equal to 
of the total degrees in the clock.
Since the clock is a circle, it contains 360 degrees.
of 360 is 60; therefore, the answer choice 7 or 11 proves to be correct.
When the hour hand is pointed to 45 minutes (or the 9th hour), the hour hand must be pointed to either 7 or 11.
This is because the angle created will span between 9 and 11 or 9 and 7, which spans "2 hours" worth of time on the clock. Given that there are a total of 12 hours of time on the clock, this means that the angle formed will be equal to
Since the clock is a circle, it contains 360 degrees.
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Which equation is the formula for chord length?
Note: 
is the radius of the circle, and 
is the angle cut by the chord.
Which equation is the formula for chord length?
Note:
The length of a chord of a circle is calculated as follows:
Chord length = 
The length of a chord of a circle is calculated as follows:
Chord length =
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The radius of a circle is 
, and the perpendicular distance from a chord to the circle center is 
. Give the chord length.
The radius of a circle is
Chord length = 
, where 
is the radius of the circle and 
is the perpendicular distance from the chord to the circle center.
Chord length = 
Chord length = 
Chord length =
Chord length =
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In the circle below, the radius is 
and the chord length is 
. Give the perpendicular distance from the chord to the circle center (d).
In the circle below, the radius is
Chord length = 
, where 
is the radius of the circle and 
is the perpendicular distance from the chord to the circle center.
Chord length = 
Chord length =
Chord length =
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