How to find the area of a circle - PSAT Math
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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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If a circle has a circumference of 16π, what would its area be if its radius were halved?
If a circle has a circumference of 16π, what would its area be if its radius were halved?
The circumference of a circle = πd where d = diameter. Therefore, this circle’s diameter must equal 16. Knowing that diameter = 2 times the radius, we can determine that the radius of this circle = 8. Halving the radius would give us a new radius of 4. To find the area of this new circle, use the formula A=πr² where r = radius. Plug in 4 for r. Area will equal 16π.
The circumference of a circle = πd where d = diameter. Therefore, this circle’s diameter must equal 16. Knowing that diameter = 2 times the radius, we can determine that the radius of this circle = 8. Halving the radius would give us a new radius of 4. To find the area of this new circle, use the formula A=πr² where r = radius. Plug in 4 for r. Area will equal 16π.
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Kate has a ring-shaped lawn which has an inner radius of 10 feet and an outer radius 25 feet. What is the area of her lawn?
Kate has a ring-shaped lawn which has an inner radius of 10 feet and an outer radius 25 feet. What is the area of her lawn?
The area of an annulus is
where 
is the radius of the larger circle, and 
is the radius of the smaller circle.
The area of an annulus is
where
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A star is inscribed in a circle with a diameter of 30, given the area of the star is 345, find the area of the shaded region, rounded to one decimal.
A star is inscribed in a circle with a diameter of 30, given the area of the star is 345, find the area of the shaded region, rounded to one decimal.
The area of the circle is (30/2)2*3.14 (π) = 706.5, since the shaded region is simply the area difference between the circle and the star, it’s 706.5-345 = 361.5
The area of the circle is (30/2)2*3.14 (π) = 706.5, since the shaded region is simply the area difference between the circle and the star, it’s 706.5-345 = 361.5
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If a circle has circumference 
, what is its area?
If a circle has circumference 
If the circumference is 
, then since 
we know 
. We further know that 
, so 
If the circumference is 
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The diameter of a circle increases by 100 percent. If the original area is 16π, what is the new area of the circle?
The diameter of a circle increases by 100 percent. If the original area is 16π, what is the new area of the circle?
The original radius would be 4, making the new radius 8 and by the area of a circle (A=π(r)2) the new area would be 64π.
The original radius would be 4, making the new radius 8 and by the area of a circle (A=π(r)2) the new area would be 64π.
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Two equal circles are cut out of a rectangular sheet of paper with the dimensions 10 by 20. The circles were made to have the greatest possible diameter. What is the approximate area of the paper after the two circles have been cut out?
Two equal circles are cut out of a rectangular sheet of paper with the dimensions 10 by 20. The circles were made to have the greatest possible diameter. What is the approximate area of the paper after the two circles have been cut out?
The length of 20 represents the diameters of both circles. Each circle has a diameter of 10 and since radius is half of the diameter, each circle has a radius of 5. The area of a circle is A = πr2 . The area of one circle is 25π. The area of both circles is 50π. The area of the rectangle is (10)(20) = 200. 200 - 50π gives you the area of the paper after the two circles have been cut out. π is about 3.14, so 200 – 50(3.14) = 43.
The length of 20 represents the diameters of both circles. Each circle has a diameter of 10 and since radius is half of the diameter, each circle has a radius of 5. The area of a circle is A = πr2 . The area of one circle is 25π. The area of both circles is 50π. The area of the rectangle is (10)(20) = 200. 200 - 50π gives you the area of the paper after the two circles have been cut out. π is about 3.14, so 200 – 50(3.14) = 43.
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A 6 by 8 rectangle is inscribed in a circle. What is the area of the circle?
A 6 by 8 rectangle is inscribed in a circle. What is the area of the circle?
The image below shows the rectangle inscribed in the circle. Dividing the rectangle into two triangles allows us to find the diameter of the circle, which is equal to the length of the line we drew. Using a2+b2= c2 we get 62 + 82 = c2. c2 = 100, so c = 10. The area of a circle is 
. Radius is half of the diameter of the circle (which we know is 10), so r = 5.
The image below shows the rectangle inscribed in the circle. Dividing the rectangle into two triangles allows us to find the diameter of the circle, which is equal to the length of the line we drew. Using a2+b2= c2 we get 62 + 82 = c2. c2 = 100, so c = 10. The area of a circle is
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A circle with a diameter of 6” sits inside a circle with a radius of 8”. What is the area of the interstitial space between the two circles?
A circle with a diameter of 6” sits inside a circle with a radius of 8”. What is the area of the interstitial space between the two circles?
The area of a circle is πr2.
The diameter of the first circle = 6” so radius of the first circle = 3” so the area = π * 32 = 9π in2
The radius of the second circle = 8” so the area = π * 82 = 64π in2
The area of the interstitial space = area of the first circle – area of the second circle.
Area = 64π in2 - 9π in2 = 55π in2
The area of a circle is πr2.
The diameter of the first circle = 6” so radius of the first circle = 3” so the area = π * 32 = 9π in2
The radius of the second circle = 8” so the area = π * 82 = 64π in2
The area of the interstitial space = area of the first circle – area of the second circle.
Area = 64π in2 - 9π in2 = 55π in2
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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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- 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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Assume π = 3.14
A man would like to put a circular whirlpool in his backyard. He would like the whirlpool to be six feet wide. His backyard is 8 feet long by 7 feet wide. By state regulation, in order to put a whirlpool in a backyard space, the space must be 1.5 times bigger than the pool. Can the man legally install the whirlpool?
Assume π = 3.14
A man would like to put a circular whirlpool in his backyard. He would like the whirlpool to be six feet wide. His backyard is 8 feet long by 7 feet wide. By state regulation, in order to put a whirlpool in a backyard space, the space must be 1.5 times bigger than the pool. Can the man legally install the whirlpool?
If you answered that the whirlpool’s area is 18.84 feet and therefore fits, you are incorrect because 18.84 is the circumference of the whirlpool, not the area.
If you answered that the area of the whirlpool is 56.52 feet, you multiplied the area of the whirlpool by 1.5 and assumed that that was the correct area, not the legal limit.
If you answered that the area of the backyard was smaller than the area of the whirlpool, you did not calculate area correctly.
And if you thought the area of the backyard was 30 feet, you found the perimeter of the backyard, not the area.
The correct answer is that the area of the whirlpool is 28.26 feet and, when multiplied by 1.5 = 42.39, which is smaller than the area of the backyard, which is 56 square feet.
If you answered that the whirlpool’s area is 18.84 feet and therefore fits, you are incorrect because 18.84 is the circumference of the whirlpool, not the area.
If you answered that the area of the whirlpool is 56.52 feet, you multiplied the area of the whirlpool by 1.5 and assumed that that was the correct area, not the legal limit.
If you answered that the area of the backyard was smaller than the area of the whirlpool, you did not calculate area correctly.
And if you thought the area of the backyard was 30 feet, you found the perimeter of the backyard, not the area.
The correct answer is that the area of the whirlpool is 28.26 feet and, when multiplied by 1.5 = 42.39, which is smaller than the area of the backyard, which is 56 square feet.
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A park wants to build a circular fountain with a walkway around it. The fountain will have a radius of 40 feet, and the walkway is to be 4 feet wide. If the walkway is to be poured at a depth of 1.5 feet, how many cubic feet of concrete must be mixed to make the walkway?
A park wants to build a circular fountain with a walkway around it. The fountain will have a radius of 40 feet, and the walkway is to be 4 feet wide. If the walkway is to be poured at a depth of 1.5 feet, how many cubic feet of concrete must be mixed to make the walkway?
The following diagram will help to explain the solution:
We are searching for the surface area of the shaded region. We can multiply this by the depth (1.5 feet) to find the total volume of this area.
The radius of the outer circle is 44 feet. Therefore its area is 442π = 1936π. The area of the inner circle is 402π = 1600π. Therefore the area of the shaded area is 1936π – 1600π = 336π. The volume is 1.5 times this, or 504π.
The following diagram will help to explain the solution:
We are searching for the surface area of the shaded region. We can multiply this by the depth (1.5 feet) to find the total volume of this area.
The radius of the outer circle is 44 feet. Therefore its area is 442π = 1936π. The area of the inner circle is 402π = 1600π. Therefore the area of the shaded area is 1936π – 1600π = 336π. The volume is 1.5 times this, or 504π.
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A square has an area of 1089 in2. If a circle is inscribed within the square, what is its area?
A square has an area of 1089 in2. If a circle is inscribed within the square, what is its area?
The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:
A = s2 or 1089 = s2. Taking the square root of both sides, we get: s = 33.
Now, based on this, we know that 2r = 33 or r = 16.5. The area of the circle is πr2 or π16.52 = 272.25π.
The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:
A = s2 or 1089 = s2. Taking the square root of both sides, we get: s = 33.
Now, based on this, we know that 2r = 33 or r = 16.5. The area of the circle is πr2 or π16.52 = 272.25π.
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A square has an area of 32 in2. If a circle is inscribed within the square, what is its area?
A square has an area of 32 in2. If a circle is inscribed within the square, what is its area?
The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:
A = s2 or 32 = s2. Taking the square root of both sides, we get: s = √32 = √(25) = 4√2.
Now, based on this, we know that 2r = 4√2 or r = 2√2. The area of the circle is πr2 or π(2√2)2 = 4 * 2π = 8π.
The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:
A = s2 or 32 = s2. Taking the square root of both sides, we get: s = √32 = √(25) = 4√2.
Now, based on this, we know that 2r = 4√2 or r = 2√2. The area of the circle is πr2 or π(2√2)2 = 4 * 2π = 8π.
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A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.25 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 44 in.?
A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.25 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 44 in.?
The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 88 inches for our problem. Its total area would be 88 * 88 or 7744 in2.
Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr2 or π * 442 = 1936π in2. Therefore, the area remaining would be 7744 – 1936π. The cost of the waste would be 0.25 * (7744 – 1936π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 4 from our subtraction. This would give us: 0.25 * 4 * (1936 – 484π). Since 0.25 is equal to 1/4, 0.25 * 4 = 1. Therefore, our final answer is: 1936 – 484π dollars.
The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 88 inches for our problem. Its total area would be 88 * 88 or 7744 in2.
Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr2 or π * 442 = 1936π in2. Therefore, the area remaining would be 7744 – 1936π. The cost of the waste would be 0.25 * (7744 – 1936π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 4 from our subtraction. This would give us: 0.25 * 4 * (1936 – 484π). Since 0.25 is equal to 1/4, 0.25 * 4 = 1. Therefore, our final answer is: 1936 – 484π dollars.
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A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.20 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 25 in.?
A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.20 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 25 in.?
The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 50 inches for our problem. Its total area would be 50 * 50 or 2500 in2.
Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr2 or π * 252 = 625π in2. Therefore, the area remaining would be 2500 - 625π. The cost of the waste would be 0.2 * (2500 – 625π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 5 from our subtraction. This would give us: 0.2 * 5 * (500 – 125π). Since 0.2 is equal to 1/5, 0.2 * 625 = 125. Therefore, our final answer is: 500 – 125π dollars.
The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 50 inches for our problem. Its total area would be 50 * 50 or 2500 in2.
Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr2 or π * 252 = 625π in2. Therefore, the area remaining would be 2500 - 625π. The cost of the waste would be 0.2 * (2500 – 625π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 5 from our subtraction. This would give us: 0.2 * 5 * (500 – 125π). Since 0.2 is equal to 1/5, 0.2 * 625 = 125. Therefore, our final answer is: 500 – 125π dollars.
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