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ISEE Upper Level Quantitative : How to find the angle of a sector

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Example Question #1 : How To Find The Angle Of A Sector

Which is the greater quantity?

(a) The degree measure of a 10-inch-long arc on a circle with radius 8 inches.

(b) The degree measure of a 12-inch-long arc on a circle with radius 10 inches.

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

(a) A circle with radius 8 inches has crircumference $C = 2\pi r = 2\pi \cdot 8 = 16\pi$ inches. An arc 10 inches long is $\frac{10}{16\pi } = \frac{5}{8\pi }$ of that circle. $\frac{5}{8\pi } \times 360 ^{\circ } = \frac{225}{\pi } ^{\circ }$ , the degree measure of this arc.

(b) A circle with radius 10 inches has crircumference $C = 2\pi r = 2\pi \cdot 10 = 20\pi$ inches. An arc 12 inches long is $\frac{12}{20\pi }=\frac{3}{5\pi }$ of that circle. $\frac{3}{5\pi } \times 360 ^{\circ } = \frac{216}{\pi } ^{\circ }$ , the degree measure of this arc.

(a) is the greater quantity.

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Example Question #1 : How To Find The Angle Of A Sector

Circle

Note: figure NOT drawn to scale

Refer to the above figure. Which is the greater quantity?

(a) $m \widehat{ACB}$

(b) $180 ^{\circ }$

Possible Answers:

(a) and (b) are equal

(It is impossible to tell from the information given

(b) is greater

(a) is greater

Correct answer:

(a) and (b) are equal

Explanation:

Since

$7^{2} + 12 ^{2} = 49 + 144 = 169 = 13 ^{2}$ ,

the triangle is a right triangle with right angle $\angle ACB$ .

$\angle ACB$ is an inscribed angle on the circle, so the arc it intercepts is a semicricle. Therefore, $\widehat{ACB}$ is also a semicircle, and it measures $180 ^{\circ }$ .

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Example Question #1 : How To Find The Angle Of A Sector

Circle

Note: Figure NOT drawn to scale

Refer to the above figure. Which is the greater quantity?

(a)

(b) 90

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

Correct answer:

(b) is greater

Explanation:

The measure of an arc intercepted by an inscribed angle of a circle is twice that of the angle. Therefore, $y = 2 \cdot 44 = 88 < 90$

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Example Question #4 : How To Find The Angle Of A Sector

$\odot A$ has twice the radius of $\odot B$ . Sector 1 is part of $\odot A$ ; Sector 2 is part of $\odot B$ ; the two sectors are equal in area.

Which is the greater quantity?

(a) Twice the degree measure of the central angle of Sector 1

(b) The degree measure of the central angle of Sector 2

Possible Answers:

(a) is greater

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

$\odot A$ has twice the radius of $\odot B$ , so $\odot A$ has four times the area of $\odot B$ . This means that for a sector of $\odot A$ to have the same area as a sector of $\odot B$ , the central angle of the latter sector must be four times that of the former sector. This makes (b) greater than (a), which is only twice that of the former sector.

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Example Question #2 : How To Find The Angle Of A Sector

Circle

Refer to the above figure. Which is the greater quantity?

(a)

(b) 55

Possible Answers:

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

The measure of an inscribed angle of a circle is one-half that of the arc it intercepts. Therefore, $x = \frac{1}{2} \cdot 110 = 55$ .

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Example Question #1 : How To Find The Angle Of A Sector

Circlesectorgeneral81

The arc-length for the shaded sector is 19.4 . What is the value of , rounded to the nearest hundredth?

Possible Answers:

81.89 ˚

74.13 ˚

127.76 ˚

18.33 ˚

29.37 ˚

Correct answer:

127.76 ˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total 360 degrees of the circle. The proportion of to 360 is the same as 19.4 to the total circumference of the circle.

The circumference of a circle is found by:

$C = 2*\pi*r$

For our data, this means:

$C = 2*8.7*\pi=17.4\pi$

Now we can solve for using the proportions:

$\frac{x}{360} = \frac{19.4}{17.4\pi}$

Cross multiply:

$17.4x\pi=6984$

Divide both sides by $17.4\pi$ :

x=127.76300259239047

Therefore, is 127.76 ˚.

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Example Question #7 : How To Find The Angle Of A Sector

Circlesectorgeneral9

The arc-length for the shaded sector is 9.31 . What is the value of , rounded to the nearest hundredth?

Possible Answers:

44.18 ˚

15.83 ˚

78.13 ˚

58.62 ˚

83.14 ˚

Correct answer:

58.62 ˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total 360 degrees of the circle. The proportion of to 360 is the same as 9.31 to the total circumference of the circle.

The circumference of a circle is found by:

$C = 2*\pi*r$

For our data, this means:

$C = 2*9.1*\pi=18.2\pi$

Now we can solve for using the proportions:

$\frac{x}{360} = \frac{9.31}{18.2\pi}$

Cross multiply:

$18.2x\pi=3351.6$

Divide both sides by $18.2\pi$ :

x=58.61798980953807

Therefore, is 58.62 ˚.

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Example Question #8 : How To Find The Angle Of A Sector

Circlesectorgeneral7.5

The area of the shaded sector in circle O is $2.81\pi$ . What is the angle measure , rounded to the nearest hundredth?

Possible Answers:

94.13 ˚

17.98 ˚

78.41 ˚

23.11 ˚

67.44 ˚

Correct answer:

17.98 ˚

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total 360 degrees of the circle. The proportion of to 360 is the same as $4.5\pi$ to the total area of the circle.

The area of a circle is found by:

$A = \pi * r^2$

For our data, this means:

$A = 7.5^2\pi = 56.25\pi$

Now we can solve for using the proportions:

$\frac{x}{360} = \frac{2.81\pi}{56.25\pi}$

Cross multiply:

$56.25x\pi=1011.6\pi$

Divide both sides by $56.25\pi$ :

x=17.984

Therefore, is 17.98 ˚.

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Example Question #9 : How To Find The Angle Of A Sector

Circlesectorgeneral6

The area of the shaded sector in circle O is $4.5\pi$ . What is the angle measure ?

Possible Answers:

120 ˚

150 ˚

Correct answer:

Explanation:

Remember that the angle for a sector or arc is found as a percentage of the total 360 degrees of the circle. The proportion of to 360 is the same as $4.5\pi$ to the total area of the circle.

The area of a circle is found by:

$A = \pi * r^2$

For our data, this means:

$A = 6^2\pi = 36\pi$

Now we can solve for using the proportions:

$\frac{x}{360} = \frac{4.5\pi}{36\pi}$

Cross multiply:

$36x\pi=1620\pi$

Divide both sides by $36\pi$ :

x=45

Therefore, is ˚.

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Example Question #1 : How To Find The Angle Of A Sector

Icecreamcone 3

Refer to the above figure, Which is the greater quantity?

(a) The area of $\bigtriangleup ABC$

(b) The area of the orange semicircle

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

$\bigtriangleup ABC$ has angles of degree measure 30 and 60; the third angle must measure 90 degrees, making $\bigtriangleup ABC$ a right triangle.

For the sake of simplicity, let BC = 1 ; the reasoning is independent of the actual length. The smaller leg of a 30-60-90 triangle has length equal to $\frac{\sqrt{3}}{3}$ times that of the longer leg; this is about

$AB = \frac{\sqrt{3}}{3} \approx \frac{1.7}{3} \approx 0.57$

The area of a right triangle is half the product of its legs, so

$A = \frac{1}{2} \cdot 1 \cdot 0.57 = 0.5 \cdot 0.57 \approx 0.285$

Also, if BC = 1 , then the orange semicircle has diameter 1 and radius $\frac{1}{2}$ . Its area can be found by substituting $r = \frac{1}{2}$ in the formula:

$A = \frac{1}{2} \cdot \pi r^{2}$

$= \frac{1}{2} \cdot \pi \left ( \frac{1}{2} \right ) ^{2}$

$= \frac{1}{2} \cdot \pi \cdot \left ( \frac{1}{4 } \right )$

$= \frac{1}{8} \cdot \pi$

$\approx 0. 125 \cdot 3.14$

$\approx 0. 3925$

The orange semicircle has a greater area than $\bigtriangleup ABC$

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ISEE Upper Level Quantitative : How to find the angle of a sector

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

Example Question #4 : How To Find The Angle Of A Sector

Example Question #2 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

Example Question #7 : How To Find The Angle Of A Sector

Example Question #8 : How To Find The Angle Of A Sector

Example Question #9 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

All ISEE Upper Level Quantitative Resources

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