Radius - PSAT Math
Card 1 of 371
Consider a circle centered at the origin with a circumference of 13pi. What is the x value when y = 3? Round your answer to the hundreths place.
Consider a circle centered at the origin with a circumference of 13pi. What is the x value when y = 3? Round your answer to the hundreths place.
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The formula for circumference of a circle is C=2pi r, so we can solve for r:
2pi r=13pi
$\frac{2pi r}{pi}$=\frac{13pi}{pi}$
2r=13
r=\frac{13}{2}$=6.5
We now know that the hypotenuse of the right triangle's length is 13.5. We can form a right triangle from the unit circle that fits the Pythagorean theorem as such:
$x^2$$+y^2$$=r^2$
Or, in this case:
$x^2$$+3^3$$=6.5^2$
$x^2$=42.26-9
$x^2$=33.25
x=$\sqrt{33.25}$=5.77
The formula for circumference of a circle is C=2pi r, so we can solve for r:
2pi r=13pi
$\frac{2pi r}{pi}$=\frac{13pi}{pi}$
2r=13
r=\frac{13}{2}$=6.5
We now know that the hypotenuse of the right triangle's length is 13.5. We can form a right triangle from the unit circle that fits the Pythagorean theorem as such:
$x^2$$+y^2$$=r^2$
Or, in this case:
$x^2$$+3^3$$=6.5^2$
$x^2$=42.26-9
$x^2$=33.25
x=$\sqrt{33.25}$=5.77
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In the figure above, rectangle ABCD has a perimeter of 40. If the shaded region is a semicircle with an area of 18π, then what is the area of the unshaded region?
In the figure above, rectangle ABCD has a perimeter of 40. If the shaded region is a semicircle with an area of 18π, then what is the area of the unshaded region?
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In order to find the area of the unshaded region, we will need to find the area of the rectangle and then subtract the area of the semicircle. However, to find the area of the rectangle, we will need to find both its length and its width. We can use the circle to find the length of the rectangle, because the length of the rectangle is equal to the diameter of the circle.
First, we can use the formula for the area of a circle in order to find the circle's radius. When we double the radius, we will have the diameter of the circle and, thus, the length of the rectangle. Then, once we have the rectangle's length, we can find its width because we know the rectangle's perimeter.
Area of a circle = πr2
Area of a semicircle = (1/2)πr2 = 18π
Divide both sides by π, then multiply both sides by 2.
r2 = 36
Take the square root.
r = 6.
The radius of the circle is 6, and therefore the diameter is 12. Keep in mind that the diameter of the circle is also equal to the length of the rectangle.
If we call the length of the rectangle l, and we call the width w, we can write the formula for the perimeter as 2l + 2w.
perimeter of rectangle = 2l + 2w
40 = 2(12) + 2w
Subtract 24 from both sides.
16 = 2w
w = 8.
Since the length of the rectangle is 12 and the width is 8, we can now find the area of the rectangle.
Area = l x w = 12(8) = 96.
Finally, to find the area of just the unshaded region, we must subtract the area of the circle, which is 18π, from the area of the rectangle.
area of unshaded region = 96 – 18π.
The answer is 96 – 18π.
In order to find the area of the unshaded region, we will need to find the area of the rectangle and then subtract the area of the semicircle. However, to find the area of the rectangle, we will need to find both its length and its width. We can use the circle to find the length of the rectangle, because the length of the rectangle is equal to the diameter of the circle.
First, we can use the formula for the area of a circle in order to find the circle's radius. When we double the radius, we will have the diameter of the circle and, thus, the length of the rectangle. Then, once we have the rectangle's length, we can find its width because we know the rectangle's perimeter.
Area of a circle = πr2
Area of a semicircle = (1/2)πr2 = 18π
Divide both sides by π, then multiply both sides by 2.
r2 = 36
Take the square root.
r = 6.
The radius of the circle is 6, and therefore the diameter is 12. Keep in mind that the diameter of the circle is also equal to the length of the rectangle.
If we call the length of the rectangle l, and we call the width w, we can write the formula for the perimeter as 2l + 2w.
perimeter of rectangle = 2l + 2w
40 = 2(12) + 2w
Subtract 24 from both sides.
16 = 2w
w = 8.
Since the length of the rectangle is 12 and the width is 8, we can now find the area of the rectangle.
Area = l x w = 12(8) = 96.
Finally, to find the area of just the unshaded region, we must subtract the area of the circle, which is 18π, from the area of the rectangle.
area of unshaded region = 96 – 18π.
The answer is 96 – 18π.
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In a large field, a circle with an area of 144_π_ square meters is drawn out. Starting at the center of the circle, a groundskeeper mows in a straight line to the circle's edge. He then turns and mows ¼ of the way around the circle before turning again and mowing another straight line back to the center. What is the length, in meters, of the path the groundskeeper mowed?
In a large field, a circle with an area of 144_π_ square meters is drawn out. Starting at the center of the circle, a groundskeeper mows in a straight line to the circle's edge. He then turns and mows ¼ of the way around the circle before turning again and mowing another straight line back to the center. What is the length, in meters, of the path the groundskeeper mowed?
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Circles have an area of πr_2, where r is the radius. If this circle has an area of 144_π, then you can solve for the radius:
πr_2 = 144_π
r 2 = 144
r =12
When the groundskeeper goes from the center of the circle to the edge, he's creating a radius, which is 12 meters.
When he travels ¼ of the way around the circle, he's traveling ¼ of the circle's circumference. A circumference is 2_πr_. For this circle, that's 24_π_ meters. One-fourth of that is 6_π_ meters.
Finally, when he goes back to the center, he's creating another radius, which is 12 meters.
In all, that's 12 meters + 6_π_ meters + 12 meters, for a total of 24 + 6_π_ meters.
Circles have an area of πr_2, where r is the radius. If this circle has an area of 144_π, then you can solve for the radius:
πr_2 = 144_π
r 2 = 144
r =12
When the groundskeeper goes from the center of the circle to the edge, he's creating a radius, which is 12 meters.
When he travels ¼ of the way around the circle, he's traveling ¼ of the circle's circumference. A circumference is 2_πr_. For this circle, that's 24_π_ meters. One-fourth of that is 6_π_ meters.
Finally, when he goes back to the center, he's creating another radius, which is 12 meters.
In all, that's 12 meters + 6_π_ meters + 12 meters, for a total of 24 + 6_π_ meters.
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A circle has an area of 36π inches. What is the radius of the circle, in inches?
A circle has an area of 36π inches. What is the radius of the circle, in inches?
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We know that the formula for the area of a circle is π_r_2. Therefore, we must set 36π equal to this formula to solve for the radius of the circle.
36π = π_r_2
36 = _r_2
6 = r
We know that the formula for the area of a circle is π_r_2. Therefore, we must set 36π equal to this formula to solve for the radius of the circle.
36π = π_r_2
36 = _r_2
6 = r
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Circle X is divided into 3 sections: A, B, and C. The 3 sections are equal in area. If the area of section C is 12π, what is the radius of the circle?
Circle X
Circle X is divided into 3 sections: A, B, and C. The 3 sections are equal in area. If the area of section C is 12π, what is the radius of the circle?
Circle X
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Find the total area of the circle, then use the area formula to find the radius.
Area of section A = section B = section C
Area of circle X = A + B + C = 12π+ 12π + 12π = 36π
Area of circle = where r is the radius of the circle
36π = πr2
36 = r2
√36 = r
6 = r
Find the total area of the circle, then use the area formula to find the radius.
Area of section A = section B = section C
Area of circle X = A + B + C = 12π+ 12π + 12π = 36π
Area of circle = where r is the radius of the circle
36π = πr2
36 = r2
√36 = r
6 = r
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The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate radius of the basketball?
The specifications of an official NBA basketball are that it must be 29.5 inches in circumference and weigh 22 ounces. What is the approximate radius of the basketball?
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To Find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get \[29.5\]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. (The information given of 22 ounces is useless)
To Find your answer, we would use the formula: C=2πr. We are given that C = 29.5. Thus we can plug in to get \[29.5\]=2πr and then multiply 2π to get 29.5=(6.28)r. Lastly, we divide both sides by 6.28 to get 4.70=r. (The information given of 22 ounces is useless)
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Two concentric circles have circumferences of 4π and 10π. What is the difference of the radii of the two circles?
Two concentric circles have circumferences of 4π and 10π. What is the difference of the radii of the two circles?
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The circumference of any circle is 2πr, where r is the radius.
Therefore:
The radius of the smaller circle with a circumference of 4π is 2 (from 2πr = 4π).
The radius of the larger circle with a circumference of 10π is 5 (from 2πr = 10π).
The difference of the two radii is 5-2 = 3.
The circumference of any circle is 2πr, where r is the radius.
Therefore:
The radius of the smaller circle with a circumference of 4π is 2 (from 2πr = 4π).
The radius of the larger circle with a circumference of 10π is 5 (from 2πr = 10π).
The difference of the two radii is 5-2 = 3.
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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?
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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 circle has an area of 
. What is the radius of the circle, in inches?
A circle has an area of
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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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A certain circle has a circumference (in units) that is two times the area (in units squared). What is the radius of the circle?
A certain circle has a circumference (in units) that is two times the area (in units squared). What is the radius of the circle?
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The formula for the area of a circle is
The formula for the circumference of a circle is
Because the circumference is twice the area, your new formula is
You can divide both sides by 
, giving you the answer: 
.
The formula for the area of a circle is
The formula for the circumference of a circle is
Because the circumference is twice the area, your new formula is
You can divide both sides by
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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?
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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?
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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?
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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.
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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 
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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?
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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?
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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?
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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?
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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?
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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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