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SAT Math : Circles

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Example Questions

Example Question #2 : How To Find The Length Of A Radius

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

Possible Answers:

Correct answer:

Explanation:

We know that the formula for the area of a circle is πr². Therefore, we must set 36π equal to this formula to solve for the radius of the circle.

36π = πr²

36 = r²

6 = r

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Example Question #2 : How To Find The Length Of A Radius

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?

Act_math_170_02

Circle X

Possible Answers:

√12

Correct answer:

Explanation:

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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Example Question #1 : How To Find The Length Of A Radius

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?

Possible Answers:

14.75 inches

3.06 inches

4.70 inches

9.39 inches

5.43 inches

Correct answer:

4.70 inches

Explanation:

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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Example Question #3 : How To Find The Length Of A Radius

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?

Possible Answers:

Correct answer:

Explanation:

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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Example Question #3 : How To Find The Length Of A Radius

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

Possible Answers:

16 inches

7 inches

24.5 inches

49 inches

14 inches

Correct answer:

7 inches

Explanation:

We know that the formula for the area of a circle is πr². Therefore, we must set 49π equal to this formula to solve for the radius of the circle.

49π = πr²

49 = r²

7 = r

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Example Question #111 : Circles

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

Possible Answers:

Correct answer:

Explanation:

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

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Example Question #361 : Plane Geometry

Chords

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

Possible Answers:

135

The question cannot be answered with the information given.

$33\frac{3}{4}$

$67\frac{1}{2}$

$16\frac{7}{8}$

Correct answer:

$33\frac{3}{4}$

Explanation:

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

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

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

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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Example Question #362 : Plane Geometry

Secant

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

Possible Answers:

The question cannot be answered with the information given.

242

198

121

Correct answer:

Explanation:

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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Example Question #61 : Radius

Give the radius of a circle with diameter fifteen yards.

Possible Answers:

$300 \textrm{ in}$

$270 \textrm{ in}$

$280 \textrm{ in}$

$240 \textrm{ in}$

$320 \textrm{ in}$

Correct answer:

$270 \textrm{ in}$

Explanation:

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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Example Question #361 : Plane Geometry

Tangents

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

Possible Answers:

102

The question cannot be answered with the information given.

Correct answer:

Explanation:

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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SAT Math : Circles

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #2 : How To Find The Length Of A Radius

Example Question #2 : How To Find The Length Of A Radius

Example Question #1 : How To Find The Length Of A Radius

Example Question #3 : How To Find The Length Of A Radius

Example Question #3 : How To Find The Length Of A Radius

Example Question #111 : Circles

Example Question #361 : Plane Geometry

Example Question #362 : Plane Geometry

Example Question #61 : Radius

Example Question #361 : Plane Geometry

All SAT Math Resources

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