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Intermediate Geometry : Sectors

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #121 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 9$ has an area of $16\pi$ $\displaystyle 16\pi$. In degrees, what is the measurement of the central angle of the circle?

Possible Answers:

$74.11^\circ$ $\displaystyle 74.11^\circ$

$81.11^\circ$ $\displaystyle 81.11^\circ$

$71.11^\circ$ $\displaystyle 71.11^\circ$

$72.11^\circ$ $\displaystyle 72.11^\circ$

Correct answer:

$71.11^\circ$ $\displaystyle 71.11^\circ$

Explanation:

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 16\pi}{\pi\times 9^2}=\frac{5760\pi}{81\pi}=71.11$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 16\pi}{\pi\times 9^2}=\frac{5760\pi}{81\pi}=71.11$

The central angle is 71.11 $\displaystyle 71.11$ degrees.

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #122 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 9$ has an area of $9\pi$ $\displaystyle 9\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$40^\circ$ $\displaystyle 40^\circ$

$25^\circ$ $\displaystyle 25^\circ$

$35^\circ$ $\displaystyle 35^\circ$

$30^\circ$ $\displaystyle 30^\circ$

Correct answer:

$40^\circ$ $\displaystyle 40^\circ$

Explanation:

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 9\pi}{\pi\times 9^2}=\frac{3240\pi}{81\pi}=40$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 9\pi}{\pi\times 9^2}=\frac{3240\pi}{81\pi}=40$

The central angle is $\displaystyle 40$ degrees.

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Example Question #123 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 9$ has an area of $2\pi$ $\displaystyle 2\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$12.89^\circ$ $\displaystyle 12.89^\circ$

$6.89^\circ$ $\displaystyle 6.89^\circ$

$10.89^\circ$ $\displaystyle 10.89^\circ$

$8.89^\circ$ $\displaystyle 8.89^\circ$

Correct answer:

$8.89^\circ$ $\displaystyle 8.89^\circ$

Explanation:

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 2\pi}{\pi\times 9^2}=\frac{720\pi}{81\pi}=8.89$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 2\pi}{\pi\times 9^2}=\frac{720\pi}{81\pi}=8.89$

The central angle is 8.89 $\displaystyle 8.89$ degrees.

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #124 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 9$ has an area of $3\pi$ $\displaystyle 3\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$16.33^\circ$ $\displaystyle 16.33^\circ$

$13.33^\circ$ $\displaystyle 13.33^\circ$

$10.33^\circ$ $\displaystyle 10.33^\circ$

$9.33^\circ$ $\displaystyle 9.33^\circ$

Correct answer:

$13.33^\circ$ $\displaystyle 13.33^\circ$

Explanation:

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 3\pi}{\pi\times 9^2}=\frac{1080\pi}{81\pi}=13.33$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 3\pi}{\pi\times 9^2}=\frac{1080\pi}{81\pi}=13.33$

The central angle is 13.33 $\displaystyle 13.33$ degrees.

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #125 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 12$ has an area of $12\pi$ $\displaystyle 12\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$30^\circ$ $\displaystyle 30^\circ$

$60^\circ$ $\displaystyle 60^\circ$

$40^\circ$ $\displaystyle 40^\circ$

$15^\circ$ $\displaystyle 15^\circ$

Correct answer:

$30^\circ$ $\displaystyle 30^\circ$

Explanation:

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 12\pi}{\pi\times 12^2}=\frac{4320\pi}{144\pi}=30$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 12\pi}{\pi\times 12^2}=\frac{4320\pi}{144\pi}=30$

The central angle is $\displaystyle 30$ degrees.

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Example Question #126 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 12$ has an area of $100\pi$ $\displaystyle 100\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$250^\circ$ $\displaystyle 250^\circ$

$190^\circ$ $\displaystyle 190^\circ$

$220^\circ$ $\displaystyle 220^\circ$

$160^\circ$ $\displaystyle 160^\circ$

Correct answer:

$250^\circ$ $\displaystyle 250^\circ$

Explanation:

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 100\pi}{\pi\times 12^2}=\frac{36000\pi}{144\pi}=250$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 100\pi}{\pi\times 12^2}=\frac{36000\pi}{144\pi}=250$

The central angle is 250 $\displaystyle 250$ degrees.

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Example Question #127 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 12$ has an area of $120\pi$ $\displaystyle 120\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$280^\circ$ $\displaystyle 280^\circ$

$250^\circ$ $\displaystyle 250^\circ$

$320^\circ$ $\displaystyle 320^\circ$

$300^\circ$ $\displaystyle 300^\circ$

Correct answer:

$300^\circ$ $\displaystyle 300^\circ$

Explanation:

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 120\pi}{\pi\times 12^2}=\frac{43200\pi}{144\pi}=300$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 120\pi}{\pi\times 12^2}=\frac{43200\pi}{144\pi}=300$

The central angle is 300 $\displaystyle 300$ degrees.

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Example Question #128 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 12$ has an area of $45\pi$ $\displaystyle 45\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$118.5^\circ$ $\displaystyle 118.5^\circ$

$106.5^\circ$ $\displaystyle 106.5^\circ$

$112.5^\circ$ $\displaystyle 112.5^\circ$

$100.5^\circ$ $\displaystyle 100.5^\circ$

Correct answer:

$112.5^\circ$ $\displaystyle 112.5^\circ$

Explanation:

Recall how to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Measurement of Central Angle}}{360}\times\pi\times r^2$

Since the question asks for the measurement of the central angle, rearrange the equation like thus:

$\text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times\text{Area of Sector}}{\pi\times r^2}$

Plug in the given information to find the measurement of the central angle.

$\text{Measurement of Central Angle}=\frac{360\times 45\pi}{\pi\times 12^2}=\frac{16200\pi}{144\pi}=112.5$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 45\pi}{\pi\times 12^2}=\frac{16200\pi}{144\pi}=112.5$

The central angle is 112.5 $\displaystyle 112.5$ degrees.

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Example Question #21 : How To Find The Angle Of A Sector

Circle z

Refer to the above figure.

True, false, or undetermined: $\angle AWB$ $\displaystyle \angle AWB$ is a right angle.

Possible Answers:

Undetermined

True

False

Correct answer:

True

Explanation:

$\angle AVB$ $\displaystyle \angle AVB$ and $\angle AWB$ $\displaystyle \angle AWB$, being inscribed angles of the same circle which intercept the same arc, are congruent angles. $\angle AVB$ $\displaystyle \angle AVB$ is right, so $\angle AWB$ $\displaystyle \angle AWB$ must also be right.

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Example Question #122 : Sectors

True or false: A $45^{\circ }$ $\displaystyle 45^{\circ }$ sector of a circle comprises one eighth of the circle.

Possible Answers:

False

True

Correct answer:

True

Explanation:

A circle includes $360^{\circ }$ $\displaystyle 360^{\circ }$ of angle measure, so a sector of measure $N^{\circ }$ $\displaystyle N^{\circ }$ is $\frac{N}{360}$ $\displaystyle \frac{N}{360}$ of the circle. Set N = 45 $\displaystyle N = 45$ - the sector is

$\frac{45}{360} = \frac{1}{8}$ $\displaystyle \frac{45}{360} = \frac{1}{8}$

of the circle. The statement is true.

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Intermediate Geometry : Sectors

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #121 : Plane Geometry

Example Question #122 : Plane Geometry

Example Question #123 : Plane Geometry

Example Question #124 : Plane Geometry

Example Question #125 : Plane Geometry

Example Question #126 : Plane Geometry

Example Question #127 : Plane Geometry

Example Question #128 : Plane Geometry

Example Question #21 : How To Find The Angle Of A Sector

Example Question #122 : Sectors

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