Circles - ACT Math
Card 0 of 1143
A 12x16 rectangle is inscribed in a circle. What is the area of the circle?
A 12x16 rectangle is inscribed in a circle. What is the area of the circle?
Explanation: Visualizing the rectangle inside the circle (corners touching the circumference of the circle and the center of the rectangle is the center of the circle) you will see that the rectangle can be divided into 8 congruent right triangles, with the hypotenuse as the radius of the circle. Calculating the radius you divide each side of the rectangle by two for the sides of each right triangle (giving 6 and 8). The hypotenuse (by pythagorean theorem or just knowing right triangle sets) the hypotenuse is give as 10. Area of a circle is given by πr2. 102 is 100, so 100π is the area.
Explanation: Visualizing the rectangle inside the circle (corners touching the circumference of the circle and the center of the rectangle is the center of the circle) you will see that the rectangle can be divided into 8 congruent right triangles, with the hypotenuse as the radius of the circle. Calculating the radius you divide each side of the rectangle by two for the sides of each right triangle (giving 6 and 8). The hypotenuse (by pythagorean theorem or just knowing right triangle sets) the hypotenuse is give as 10. Area of a circle is given by πr2. 102 is 100, so 100π is the area.
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If the radius of a circle is tripled, and the new area is 144π what was the diameter of the original circle?
If the radius of a circle is tripled, and the new area is 144π what was the diameter of the original circle?
The area of a circle is A=πr2. Since the radius was tripled 144π =π(3r)2. Divide by π and then take the square root of both sides of the equal sign to get 12=3r, and then r=4. The diameter (d) is equal to twice the radius so d= 2(4) = 8.
The area of a circle is A=πr2. Since the radius was tripled 144π =π(3r)2. Divide by π and then take the square root of both sides of the equal sign to get 12=3r, and then r=4. The diameter (d) is equal to twice the radius so d= 2(4) = 8.
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If the equation of a circle is (x – 7)2 + (y + 1)2 = 81, what is the area of the circle?
If the equation of a circle is (x – 7)2 + (y + 1)2 = 81, what is the area of the circle?
The equation is already in a circle equation, and the right side of the equation stands for r2 → r2 = 81 and r = 9
The area of a circle is πr2, so the area of this circle is 81π.
The equation is already in a circle equation, and the right side of the equation stands for r2 → r2 = 81 and r = 9
The area of a circle is πr2, so the area of this circle is 81π.
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If the radius of Circle A is three times the radius of Circle B, what is the ratio of the area of Circle A to the area of Circle B?
If the radius of Circle A is three times the radius of Circle B, what is the ratio of the area of Circle A to the area of Circle B?
We know that the equation for the area of a circle is π r2. To solve this problem, we pick radii for Circles A and B, making sure that Circle A’s radius is three times Circle B’s radius, as the problem specifies. Then we will divide the resulting areas of the two circles. For example, if we say that Circle A has radius 6 and Circle B has radius 2, then the ratio of the area of Circle A to B is: (π 62)/(π 22) = 36π/4π. From here, the π's cancel out, leaving 36/4 = 9.
We know that the equation for the area of a circle is π r2. To solve this problem, we pick radii for Circles A and B, making sure that Circle A’s radius is three times Circle B’s radius, as the problem specifies. Then we will divide the resulting areas of the two circles. For example, if we say that Circle A has radius 6 and Circle B has radius 2, then the ratio of the area of Circle A to B is: (π 62)/(π 22) = 36π/4π. From here, the π's cancel out, leaving 36/4 = 9.
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It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?
It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?
At four o’clock the minute hand is on the 12 and the hour hand is on the 4. The angle formed is 4/12 of the total number of degrees in a circle, 360.
4/12 * 360 = 120 degrees
At four o’clock the minute hand is on the 12 and the hour hand is on the 4. The angle formed is 4/12 of the total number of degrees in a circle, 360.
4/12 * 360 = 120 degrees
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- A circle is inscribed inside a 10 by 10 square. What is the area of the circle?
- A circle is inscribed inside a 10 by 10 square. What is the area of the circle?
Area of a circle = A = πr2
R = 1/2d = ½(10) = 5
A = 52π = 25π
Area of a circle = A = πr2
R = 1/2d = ½(10) = 5
A = 52π = 25π
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A circle with center (8, **–**5) is tangent to the y-axis in the standard (x,y) coordinate plane. What is the radius of this circle?
A circle with center (8, **–**5) is tangent to the y-axis in the standard (x,y) coordinate plane. What is the radius of this circle?
For the circle to be tangent to the y-axis, it must have its outer edge on the axis. The center is 8 units from the edge.
For the circle to be tangent to the y-axis, it must have its outer edge on the axis. The center is 8 units from the edge.
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A sector of an angle is bounded the major arc of 
. What percent of the circle does this make up?
A sector of an angle is bounded the major arc of
The only information provided is that the sector is 
. When finding a percent, it's helpful to remember that percents are a clean way to show fractions—that is portions—of a whole. In order to solve for what percent the sector makes of the entire circle, we need to find out what fractionthe sector makes up of the entire circle.
Keep in mind that a circle measures 
degrees. This means that the sector is 
of the total 
. This can be written out as:
This can be simplified to:
Note that the degrees signs are gone. This is because when you divide degrees by degrees, the units cancel out! This simplified fraction means that 
was three-fourths of the entire circle. Now, this fraction may be turned into a percent.
In order to go from fractions to percents, it's helpful to keep in mind that decimals are the important intermediate step. That is, as soon as a fraction can be turned into decimal form, it can easily be converted into a percent by multiplying by 
.
That is, the sector makes up 
of the entire circle.
The only information provided is that the sector is
Keep in mind that a circle measures
This can be simplified to:
Note that the degrees signs are gone. This is because when you divide degrees by degrees, the units cancel out! This simplified fraction means that
In order to go from fractions to percents, it's helpful to keep in mind that decimals are the important intermediate step. That is, as soon as a fraction can be turned into decimal form, it can easily be converted into a percent by multiplying by
That is, the sector makes up
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Using the 8 hour analog clock from question 1 (an analog clock with 8 evenly spaced numbers on its face, with 8 where 12 normally is), what is the angle between the hands at 1:30? (Note: calculate the smaller angle, the one going between the hour and minute hand in a clockwise direction.)
Using the 8 hour analog clock from question 1 (an analog clock with 8 evenly spaced numbers on its face, with 8 where 12 normally is), what is the angle between the hands at 1:30? (Note: calculate the smaller angle, the one going between the hour and minute hand in a clockwise direction.)
Because it's an 8 hour clock, each section of the clock has an angle of 45 degrees due to the fact that 
.
When the clock reads 1:30 the hour hand is halfway in between the 1 and the 2, and the minute hand is on the 4 (at the bottom of the clock). Therefore, between the hour hand and the "2" on the clock there are 
degrees and between the 2 and the 4 there are 
degrees. Finally, 
Because it's an 8 hour clock, each section of the clock has an angle of 45 degrees due to the fact that
When the clock reads 1:30 the hour hand is halfway in between the 1 and the 2, and the minute hand is on the 4 (at the bottom of the clock). Therefore, between the hour hand and the "2" on the clock there are
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In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?
In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?
Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.
Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.
Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.
Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.
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A circle has an area of 
. What is the radius of the circle, in inches?
A circle has an area of
We know that the formula for the area of a circle is πr_2. Therefore, we must set 49_π equal to this formula to solve for the radius of the circle.
49_π_ = _πr_2
49 = _r_2
7 = r
We know that the formula for the area of a circle is πr_2. Therefore, we must set 49_π equal to this formula to solve for the radius of the circle.
49_π_ = _πr_2
49 = _r_2
7 = r
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How many degrees are in each hour-long section of an analog clock with 8 equally spaced numbers on the face?
How many degrees are in each hour-long section of an analog clock with 8 equally spaced numbers on the face?
If creating a picture helps, draw a circle and place 8 at the top where 12 normally is. Then put 2, 4, and 6 at the positions of 3, 6, and 9 on a normal 12-hour analog clock. 1, 3, 5, and 7 go halfway in between each even number.
Now, because each section is equally spaced, and because there are 8 sections we simply divide the total number of degrees in a circle (
) by the number of sections (8). Thus:
If creating a picture helps, draw a circle and place 8 at the top where 12 normally is. Then put 2, 4, and 6 at the positions of 3, 6, and 9 on a normal 12-hour analog clock. 1, 3, 5, and 7 go halfway in between each even number.
Now, because each section is equally spaced, and because there are 8 sections we simply divide the total number of degrees in a circle (
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In the figure below, 
and 
are radii of the circle. If the radius of the circle is 8 units, how long, in units, is chord 
?
In the figure below,
The radii and the chord form right triangle 
. Use the Pythagorean Theorem to find the length of 
.
The radii and the chord form right triangle
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In the circle with center A, shown below, the length of radius 
is 4 mm. What is the length of chord 
?
In the circle with center A, shown below, the length of radius 
In order to solve this question, we need to notice that the radii of the circle and chord 
form a right triangle with 
as the hypotenuse. We can use the Pythagorean theorem to find chord 
.
In order to solve this question, we need to notice that the radii of the circle and chord
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A circle has a maximum chord length of 
inches. What is its radius in inches?
A circle has a maximum chord length of
Keep in mind that the largest possible chord of a circle is its diameter.
This means that the diameter of the circle is 
inches, which means an 
-inch radius.
Remember that 
,
Keep in mind that the largest possible chord of a circle is its diameter.
This means that the diameter of the circle is
Remember that
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What is the measure, in degrees, of the acute angle formed by the hands of a 12-hour clock that reads exactly 3:10?
What is the measure, in degrees, of the acute angle formed by the hands of a 12-hour clock that reads exactly 3:10?
The entire clock measures 360°. As the clock is divided into 12 sections, the distance between each number is equivalent to 30° (360/12). The distance between the 2 and the 3 on the clock is 30°. One has to account, however, for the 10 minutes that have passed. 10 minutes is 1/6 of an hour so the hour hand has also moved 1/6 of the distance between the 3 and the 4, which adds 5° (1/6 of 30°). The total measure of the angle, therefore, is 35°.
The entire clock measures 360°. As the clock is divided into 12 sections, the distance between each number is equivalent to 30° (360/12). The distance between the 2 and the 3 on the clock is 30°. One has to account, however, for the 10 minutes that have passed. 10 minutes is 1/6 of an hour so the hour hand has also moved 1/6 of the distance between the 3 and the 4, which adds 5° (1/6 of 30°). The total measure of the angle, therefore, is 35°.
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What is the measure of the smaller angle formed by the hands of an analog watch if the hour hand is on the 10 and the minute hand is on the 2?
What is the measure of the smaller angle formed by the hands of an analog watch if the hour hand is on the 10 and the minute hand is on the 2?
A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).
A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).
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What is the angle between the hands of a standard 12-hour digital clock when it is 8:15? (note, give the smaller of the two angles, the one between the hands going clockwise
What is the angle between the hands of a standard 12-hour digital clock when it is 8:15? (note, give the smaller of the two angles, the one between the hands going clockwise
When the clock reads 8:15 the minute hand is on the 3 and the hour hand is just past the 8.
Each section of the clock is
degrees.
From 3 to 8 then there are 150 degrees. However, the hour hand has moved a quarter of the way between the 8 and the 9, or a quarter of 30 degrees. 
and so 
When the clock reads 8:15 the minute hand is on the 3 and the hour hand is just past the 8.
Each section of the clock is
From 3 to 8 then there are 150 degrees. However, the hour hand has moved a quarter of the way between the 8 and the 9, or a quarter of 30 degrees.
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What is the measure of the angle between the hands of a clock at 
? (compute the angle going clockwise from the hour hand to the minute hand)
What is the measure of the angle between the hands of a clock at
Each section of the clock is 
, and by 
the hour hand has gone three quarters of the way between the 
and the 
. Thus there are 
between the hour hand and the 
numeral. The minute hand is on the 
, and there are 
between the 
and the 
. So in total there are 
between the hands
Each section of the clock is
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On a 
analog clock (there is a 
where the 
normally is, a 
in the normal position of the 
, with 
where the 
is on a standard 
analog, the 
where the 
is on a standard analog, and 
and 
are at the spots normally occupied by 
and 
respectively), what is the angle between the hands when the clock reads 
?
On a
The number of degrees between each numeral on the clock face is equal to the number of degrees in a circle divided by the number of sections:
At 
the 
hand has gone 
way through the 
between the 
and the 
. Thus there are only 
left between it and the 3. There are 120 degrees between the 3 and the 5, where the minute hand is, so the total amount of degrees between the hands is:
The number of degrees between each numeral on the clock face is equal to the number of degrees in a circle divided by the number of sections:
At
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