Circles - ACT Math
Card 0 of 1143
Assuming the clock hands extend all the way to the edge of the clock, what is the distance between the clock hands of an analog clock with a radius of 6 inches if the time is 2:30? Reduce any fractions in your answer and leave your answer in terms of 
Assuming the clock hands extend all the way to the edge of the clock, what is the distance between the clock hands of an analog clock with a radius of 6 inches if the time is 2:30? Reduce any fractions in your answer and leave your answer in terms of
To find the distance between the clock hands, first find the angle between the clock hands and use that as your central angle to find the porportion of the circumference.
The angle between the hands is 
:
To find the distance between the clock hands, first find the angle between the clock hands and use that as your central angle to find the porportion of the circumference.
The angle between the hands is
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What is the measure of the largerangle formed by the hands of a clock at 
?
What is the measure of the largerangle formed by the hands of a clock at
Like any circle, a clock contains a total of 
. Because the clock face is divided into 
equal parts, we can find the number of degrees between each number by doing 
. At 5:00 the hour hand will be at 5 and the minute hand will be at 12. Using what we just figured out, we can see that there is an angle of 
between the two hands. We are looking for the larger angle, however, so we must now do 
.
Like any circle, a clock contains a total of
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A church commissioned a glassmaker to make a circular stained glass window that was two feet in diameter. What is the area of the stained glass window, to the nearest square foot?
A church commissioned a glassmaker to make a circular stained glass window that was two feet in diameter. What is the area of the stained glass window, to the nearest square foot?
To answer this question, we need to find the area of a circle.
To find the area of a circle, we will use the following equation:
, where 
is the radius.
We are given the diameter of the circle, which is 
. To find the radius of a circle, we divide the diameter by two. So, for this data:
So, our radius is 
.
We can now plug our radius into our equation for the area of a circle:
The answer asks us what the area is to the nearest square foot. Therefore, we must round our answer to the nearest whole number. To round, we will round up if the digit right before your desired place value is 5, 6, 7, 8, or 9, and we will round down if it is 0, 1, 2, 3, or 4. Because we want the nearest square foot, we will look at the tenths place to aid in our rounding. Because there is a 1 in the tenths place, we will round down. So:
Therefore, our answer is 
.
To answer this question, we need to find the area of a circle.
To find the area of a circle, we will use the following equation:
We are given the diameter of the circle, which is
So, our radius is
We can now plug our radius into our equation for the area of a circle:
The answer asks us what the area is to the nearest square foot. Therefore, we must round our answer to the nearest whole number. To round, we will round up if the digit right before your desired place value is 5, 6, 7, 8, or 9, and we will round down if it is 0, 1, 2, 3, or 4. Because we want the nearest square foot, we will look at the tenths place to aid in our rounding. Because there is a 1 in the tenths place, we will round down. So:
Therefore, our answer is
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An outdoor clock with a face diameter of 
displays the current time as 
. Assuming each hand reaches to the edge of the clock face, what is the distance, in feet, of the minor arc between the hour and minute hand? Leave 
in your answer. Assume a 
display (not a military clock).
An outdoor clock with a face diameter of
To start, we must calculate the circumference along the clock face. Since we are required to leave 
in our answer, this is as easy as following our equation for circumference:
Next, we must figure out the positions of the hands. At 
, the clock's minute hand would be exactly on the 
, and the hour hand would be 
of the way between the 
and the 
. Therefore, the total distance 
in terms of hour markings is 
.
To find the distance between adjacent numbers, divide the circumference by 
:
, where 
is the distance in inches between adjacent hours on the clock face.
Finally, multiply this distance by the number of hours between the hands.
.
Thus, the hands are 
feet apart.
To start, we must calculate the circumference along the clock face. Since we are required to leave
Next, we must figure out the positions of the hands. At
To find the distance between adjacent numbers, divide the circumference by
Finally, multiply this distance by the number of hours between the hands.
Thus, the hands are
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An watch with a face radius of 
displays the current time as 
. Assuming each hand reaches to the edge of the clock face, what is the distance, in 
, of the minor arc between the hour and minute hand? Leave in your answer. Assume a 
display (not a military clock).
An watch with a face radius of
To start, we must calculate the circumference along the clock face. Since we are required to leave in our answer, this is as easy as following our equation for circumference:
Next, we must figure out the positions of the hands. At 
, the clock's minute hand would be exactly on the 
, and the hour hand would be 
of the way between the 
and the 
. Therefore, the total distance in terms of hour markings is 
.
To find the distance between adjacent numbers, divide the circumference by :
, where is the distance in inches between adjacent hours on the clock face.
Finally, multiply this distance by the number of hours between the hands.
.
Thus, the hands are 
apart.
To start, we must calculate the circumference along the clock face. Since we are required to leave
Next, we must figure out the positions of the hands. At
To find the distance between adjacent numbers, divide the circumference by
Finally, multiply this distance by the number of hours between the hands.
Thus, the hands are
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An military (
) clock with a face diameter of 
displays the current time as 
. Assuming each hand reaches to the edge of the clock face, what is the distance, in feet, of the minor arc between the hour and minute hand? Leave in your answer.
An military (
To start, we must calculate the circumference along the clock face. Since we are required to leave in our answer, this is as easy as following our equation for circumference:
Next, we must figure out the positions of the hands. For this problem, the clock has 24 evenly spaced numbers instead of 12, so we must cut the distance between each number mark in half compared to a normal clock face. At 
, the clock's minute hand would be exactly on the 
, and the hour hand would be 
of the way between the 
and the 
(remember, there are still only 
seconds in a minute even on a 24-hour clock. Therefore, the total distance in terms of hour markings is 
.
To find the distance between adjacent numbers, divide the circumference by 
:
, where is the distance in feet between adjacent hours on the clock face.
Finally, multiply this distance by the number of hours between the hands.
.
Thus, the hands are 
feet apart.
To start, we must calculate the circumference along the clock face. Since we are required to leave
Next, we must figure out the positions of the hands. For this problem, the clock has 24 evenly spaced numbers instead of 12, so we must cut the distance between each number mark in half compared to a normal clock face. At
To find the distance between adjacent numbers, divide the circumference by
Finally, multiply this distance by the number of hours between the hands.
Thus, the hands are
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Find the distance between the hour and minute hand of a clock at 2:00 with a hand length of 
.
Find the distance between the hour and minute hand of a clock at 2:00 with a hand length of
To find the distance, first you need to find circumference. Thus,
Then, multiply the fraction of the clock they cover. Thus,
To find the distance, first you need to find circumference. Thus,
Then, multiply the fraction of the clock they cover. Thus,
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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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Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?
For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.
The equation for the area of a circle is A = πr2.
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The perimeter of a circle is 36 π. What is the diameter of the circle?
The perimeter of a circle is 36 π. What is the diameter of the circle?
The perimeter of a circle = 2 πr = πd
Therefore d = 36
The perimeter of a circle = 2 πr = πd
Therefore d = 36
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If a circle has an area of 
, what is the diameter of the circle?
If a circle has an area of
1. Use the area to find the radius:
2. Use the radius to find the diameter:
1. Use the area to find the radius:
2. Use the radius to find the diameter:
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In a group of students, it was decided that a pizza would be divided according to its crust size. Every student wanted 3 inches of crust (measured from the outermost point of the pizza). If the pizza in question had a diameter of 14 inches, what percentage of the pizza was wasted by this manner of cutting the pizza? Round to the nearest hundredth.
In a group of students, it was decided that a pizza would be divided according to its crust size. Every student wanted 3 inches of crust (measured from the outermost point of the pizza). If the pizza in question had a diameter of 14 inches, what percentage of the pizza was wasted by this manner of cutting the pizza? Round to the nearest hundredth.
What we are looking at is a way of dividing the pizza according to arc lengths of the crust. Thus, we need to know the total circumference first. Since the diameter is 
, we know that the circumference is 
. Now, we want to ask how many ways we can divide up the pizza into pieces of 
inch crust. This is:
or approximately 
pieces.
What you need to do is take this amount and subtract off 
. This is the amount of crust that is wasted. You can then merely divide it by the original amount of divisions:
(You do not need to work in exact area or length. These relative values work fine.)
This is about 
of the pizza that is wasted.
What we are looking at is a way of dividing the pizza according to arc lengths of the crust. Thus, we need to know the total circumference first. Since the diameter is
What you need to do is take this amount and subtract off
(You do not need to work in exact area or length. These relative values work fine.)
This is about
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A circle has an area of 
. What is its diameter?
A circle has an area of
To solve a question like this, first remember that the area of a circle is defined as:
For your data, this is:
To solve for 
, first divide both sides by 
. Then take the square root of both sides. Thus you get:
The diameter of the circle is just double that:
Rounding to the nearest hundredth, you get 
.
To solve a question like this, first remember that the area of a circle is defined as:
For your data, this is:
To solve for
The diameter of the circle is just double that:
Rounding to the nearest hundredth, you get
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What is the diameter of a semi-circle that has an area of 
?
What is the diameter of a semi-circle that has an area of
To begin, be very careful to note that the question asks about a semi-circle—not a complete circle! This means that a complete circle composed out of two of these semi-circles would have an area of 
. Now, from this, we can use our area formula, which is:
For our data, this is:
Solving for 
, we get:
This can be simplified to:
The diameter is 
, which is 
or 
.
To begin, be very careful to note that the question asks about a semi-circle—not a complete circle! This means that a complete circle composed out of two of these semi-circles would have an area of
For our data, this is:
Solving for
This can be simplified to:
The diameter is
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A circle has an area of 
. What is the diameter of the circle?
A circle has an area of
The equation for the area of a circle is 
, which in this case equals 
. Therefore, 
The only thing squared that equals an integer (which is not a perfect root) is that number under a square root. Therefore, 
. Since diameter is twice the radius, 
The equation for the area of a circle is
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Find the diameter given the radius is 
.
Find the diameter given the radius is
Diameter is simply twice the radius. Therefore, 
.
Diameter is simply twice the radius. Therefore,
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Find the length of the diameter of a circle given the area is 
.
Find the length of the diameter of a circle given the area is
To solve, simply use the formula for the area of a circle, solve for r, and multiply by 2 to get the diameter. Thus,
To solve, simply use the formula for the area of a circle, solve for r, and multiply by 2 to get the diameter. Thus,
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A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?
A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?
Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.
Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.
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