How to find circumference - Math
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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?
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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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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
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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
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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.
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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 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?
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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?
Tap to reveal answer
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)
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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)
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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.
Tap to reveal answer
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
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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
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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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A circle with an area of 13_π_ in2 is centered at point C. What is the circumference of this circle?
A circle with an area of 13_π_ in2 is centered at point C. What is the circumference of this circle?
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The formula for the area of a circle is A = _πr_2.
We are given the area, and by substitution we know that 13_π_ = _πr_2.
We divide out the π and are left with 13 = _r_2.
We take the square root of r to find that r = √13.
We find the circumference of the circle with the formula C = 2_πr_.
We then plug in our values to find C = 2√13_π_.
The formula for the area of a circle is A = _πr_2.
We are given the area, and by substitution we know that 13_π_ = _πr_2.
We divide out the π and are left with 13 = _r_2.
We take the square root of r to find that r = √13.
We find the circumference of the circle with the formula C = 2_πr_.
We then plug in our values to find C = 2√13_π_.
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A 6 by 8 rectangle is inscribed in a circle. What is the circumference of the circle?
A 6 by 8 rectangle is inscribed in a circle. What is the circumference of the circle?
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First you must draw the diagram. The diagonal of the rectangle is also the diameter of the circle. The diagonal is the hypotenuse of a multiple of 2 of a 3,4,5 triangle, and therefore is 10.
Circumference = π * diameter = 10_π_.
First you must draw the diagram. The diagonal of the rectangle is also the diameter of the circle. The diagonal is the hypotenuse of a multiple of 2 of a 3,4,5 triangle, and therefore is 10.
Circumference = π * diameter = 10_π_.
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Ashley has a square room in her apartment that measures 81 square feet. What is the circumference of the largest circular area rug that she can fit in the space?
Ashley has a square room in her apartment that measures 81 square feet. What is the circumference of the largest circular area rug that she can fit in the space?
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In order to solve this question, first calculate the length of each side of the room.
The length of each side of the room is also equal to the length of the diameter of the largest circular rug that can fit in the room. Since 
, the circumference is simply
In order to solve this question, first calculate the length of each side of the room.
The length of each side of the room is also equal to the length of the diameter of the largest circular rug that can fit in the room. Since 
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A gardener wants to build a fence around their garden shown below. How many feet of fencing will they need, if the length of the rectangular side is 12 and the width is 8?
A gardener wants to build a fence around their garden shown below. How many feet of fencing will they need, if the length of the rectangular side is 12 and the width is 8?
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The shape of the garden consists of a rectangle and two semi-circles. Since they are building a fence we need to find the perimeter. The perimeter of the length of the rectangle is 24. The perimeter or circumference of the circle can be found using the equation C=2π(r), where r= the radius of the circle. Since we have two semi-circles we can find the circumference of one whole circle with a radius of 4, which would be 8π.
The shape of the garden consists of a rectangle and two semi-circles. Since they are building a fence we need to find the perimeter. The perimeter of the length of the rectangle is 24. The perimeter or circumference of the circle can be found using the equation C=2π(r), where r= the radius of the circle. Since we have two semi-circles we can find the circumference of one whole circle with a radius of 4, which would be 8π.
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If a circle has an area of 
, what is the circumference of the circle?
If a circle has an area of
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The formula for the area of a circle is πr2. For this particular circle, the area is 81π, so 81π = πr2. Divide both sides by π and we are left with r2=81. Take the square root of both sides to find r=9. The formula for the circumference of the circle is 2πr = 2π(9) = 18π. The correct answer is 18π.
The formula for the area of a circle is πr2. For this particular circle, the area is 81π, so 81π = πr2. Divide both sides by π and we are left with r2=81. Take the square root of both sides to find r=9. The formula for the circumference of the circle is 2πr = 2π(9) = 18π. The correct answer is 18π.
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A car tire has a radius of 18 inches. When the tire has made 200 revolutions, how far has the car gone in feet?
A car tire has a radius of 18 inches. When the tire has made 200 revolutions, how far has the car gone in feet?
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If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.
If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.
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A circle has the equation below. What is the circumference of the circle?
(x – 2)2 + (y + 3)2 = 9
A circle has the equation below. What is the circumference of the circle?
(x – 2)2 + (y + 3)2 = 9
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The radius is 3. Yielding a circumference of 
.
The radius is 3. Yielding a circumference of
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The diameter of a circle is defined by the two points (2,5) and (4,6). What is the circumference of this circle?
The diameter of a circle is defined by the two points (2,5) and (4,6). What is the circumference of this circle?
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We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)
For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5
If d = √5, the circumference of our circle is πd, or π√5.
We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)
For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5
If d = √5, the circumference of our circle is πd, or π√5.
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A circle has radius 
. What is the circumference, rounded to the nearest tenth?
A circle has radius
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Circumference is given by the equation 
. We can use this equation with the given radius, 4.2, to solve for the circumference.
Circumference is given by the equation
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