How to find the length of a radius - SAT Math

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Question

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?

Answer

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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Question

A circle has an area of 36π inches. What is the radius of the circle, in inches?

Answer

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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Question

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

Answer

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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Question

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?

Answer

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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Question

Two concentric circles have circumferences of 4π and 10π. What is the difference of the radii of the two circles?

Answer

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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Question

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?

Answer

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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Question

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?

Answer

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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Question

A circle has an area of $49\pi \ in^{2}$ . What is the radius of the circle, in inches?

Answer

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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Question

Consider a circle centered at the origin with a circumference of $13\pi$ . What is the x value when y = 3? Round your answer to the hundreths place.

Answer

The formula for circumference of a circle is $C=2\pi r$ , so we can solve for r:

$2\pi r=13\pi$

$\frac{2\pi r}{\pi}=\frac{13\pi}{\pi}$

$2r=13$

$r=\frac{13}{2}=6.5$

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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Question

Find the radius of a circle given the diameter is 24.

Answer

To solve, simply realize that the radius is half the diameter. Thus, our answer is 12.

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Question

In the above diagram, $\overarc {TU}$ has length $30 \pi$ . Give the radius of the circle to the nearest whole number.

Answer

Call $t = m \overarc {TU}$ . The measure of the corresponding major arc is $360 ^{\circ } - m \overarc {TU}= 360 ^{\circ } -t$

If two tangents are drawn to a circle from a point outside the circle, the measure of the angle they form is equal to half the difference of the measures of their intercepted arcs; therefore

$\frac{1}{2} \left (360 ^{\circ } -t -t \right ) = m \angle BNT$

Substituting:

$\frac{1}{2} \left (360 ^{\circ } -2t\right ) = 66 ^{\circ }$

$180 ^{\circ } -t = 66 ^{\circ }$

$180 ^{\circ } -t + t - 66 ^{\circ } = 66 ^{\circ } + t - 66 ^{\circ }$

$114 ^{\circ } = t$

Therefore, $m \overarc {TU} = 114 ^{\circ }$ . Since $\overarc {TU}$ has length $30 \pi$ , it follows that if is the circumference of the circle,

$\frac{C}{360^{\circ }} = \frac{30 \pi }{m \overarc{TU}}$

$\frac{C}{360^{\circ }} = \frac{30 \pi }{114^{\circ }}$

$C= \frac{30 \pi }{114^{\circ }} \cdot 360^{\circ }$

$C = 94\frac{14}{19} \pi$

Divide the circumference by $2 \pi$ to obtain the radius:

$r = \frac{C}{2 \pi} = \frac{94 \frac{14}{19}\pi }{2 \pi} = 47 \frac{7}{19}$ .

This makes 47 the correct choice.

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Question

In the above diagram, $\overarc {AD}$ and $\overarc {BC}$ have lengths $9 \pi$ and $18\pi$ , respectively. Give the radius of the circle.

Answer

If two chords of a circle intersect, the measure of the angles they form is equal to half the sum of the measures of the angles they intercept that is,

$\frac{1}{2} \left ( m \overarc {AD}+ m \overarc {BC} \right ) = m \angle AXD$

The ratio of the measures of arcs on the same circle is equal to that of their lengths, so

$\frac{ m \overarc {BC} }{m \overarc {AD} } = \frac{18 \pi}{9 \pi} = 2$

and

$m \overarc {BC} = 2 \cdot m \overarc {AD}$

Substituting:

$\frac{1}{2} \left ( m \overarc {AD}+ 2 \cdot m \overarc {AD} \right ) =72^{\circ }$

$\frac{1}{2} \left ( 3 \cdot m \overarc {AD} \right ) =72^{\circ }$

$\frac{3}{2} \cdot m \overarc {AD} =72^{\circ }$

$\frac{2} {3}\cdot \frac{3}{2} \cdot m \overarc {AD} =\frac{2} {3}\cdot 72^{\circ }$

$m \overarc {AD} =48^{\circ }$

if is the circumference of the circle, and the length of the arc $\overarc {AD}$ is ,

$l = \frac{m \overarc{AC}}{360 ^{\circ }} \cdot C$

$\overarc {AD}$ has length $9 \pi$ and measure $48^{\circ }$ so

$9 \pi = \frac{48^{\circ }}{360 ^{\circ }} \cdot C$

or

$9 \pi = \frac{2 }{15 } \cdot C$

Since, if the radius is , $C = 2 \pi r$ ,

$9 \pi = \frac{2 }{15 } \cdot 2 \pi r$

$9 \pi = \frac{4 \pi }{15 } r$

Solve for :

$\frac{4 \pi }{15 } r = 9 \pi$

$\frac{15 } {4 \pi } \cdot \frac{4 \pi }{15 } r =\frac{15 } {4 \pi } \cdot 9 \pi$

$r =\frac{135} {4 } = 33\frac{3}{4}$

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Question

In the above diagram, $\overarc {AT}$ and $\overarc {BT}$ have lengths $20 \pi$ and $75 \pi$ , respectively. Give the radius of the circle.

Answer

The ratio of the degree measures of the arcs is the same as that of their lengths. Therefore,

$\frac{m \overarc {BT}}{m \overarc {AT}} = \frac{75 \pi}{20 \pi} = \frac{15}{4}$

and

$m \overarc {BT}= \frac{15}{4} \cdot m \overarc {AT}$

If a secant and a tangent are drawn to a circle from a point outside the circle, the measure of the angle they form is equal to half the difference of the measures of their intercepted arcs; therefore,

$\frac{1}{2} \left ( m\overarc {BT} - m \overarc {AT} \right ) = m \angle BNT$

Substituting:

$\frac{1}{2} \left ( m\overarc {BT} - m \overarc {AT} \right ) = 50 ^{\circ }$

$\frac{1}{2} \left ( \frac{15}{4} \cdot m \overarc {AT} - m \overarc {AT} \right ) = 50 ^{\circ }$

$\frac{11}{8} \cdot m \overarc {AT} = 50 ^{\circ }$

$\frac{8}{11} \cdot \frac{11}{8} \cdot m \overarc {AT} =\frac{8}{11} \cdot 50 ^{\circ }$

$m \overarc {AT} = 36 \frac{4}{11}^{\circ }$

Since $\overarc {AT}$ has length $20 \pi$ , then, if we let be the circumference of the circle,

$\frac{C}{360^{\circ }} = \frac{20 \pi} {36\frac{4}{11}^{\circ } }$

$\frac{C}{360^{\circ }} \cdot 360^{\circ }= \frac{20 \pi} {36\frac{4}{11}^{\circ } }\cdot 360^{\circ }$

$C=198 \pi$

Divide the circumference by $2 \pi$ to obtain the radius:

$r = \frac{C}{2 \pi} = \frac{198 \pi }{2 \pi} = 99$

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Question

Give the radius of a circle with diameter fifteen yards.

Answer

Convert fifteen yards to inches by multiplying by 36:

$15 \textrm{ yd} \times 36 \textrm{ in/yd} = 540 \textrm{ in}$

The radius of a circle is one half its diameter, so multiply this by $\frac{1}{2}$ :

$\frac{1}{2} \times 540 \textrm{ in} = 270 \textrm{ in}$

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