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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #111 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 16$ has an area of $8\pi$ $\displaystyle 8\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$15.50^\circ$ $\displaystyle 15.50^\circ$

$11.25^\circ$ $\displaystyle 11.25^\circ$

$10.50^\circ$ $\displaystyle 10.50^\circ$

$9.75^\circ$ $\displaystyle 9.75^\circ$

Correct answer:

$11.25^\circ$ $\displaystyle 11.25^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times 8\pi}{\pi\times 16^2}=\frac{2880\pi}{256\pi}=11.25$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 8\pi}{\pi\times 16^2}=\frac{2880\pi}{256\pi}=11.25$

The central angle is 11.25 $\displaystyle 11.25$ degrees.

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Example Question #112 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 20$ has a area of $50\pi$ $\displaystyle 50\pi$. In degrees, what is the measurement of the central angle in the sector?

Possible Answers:

$39^\circ$ $\displaystyle 39^\circ$

$42^\circ$ $\displaystyle 42^\circ$

$45^\circ$ $\displaystyle 45^\circ$

$36^\circ$ $\displaystyle 36^\circ$

Correct answer:

$45^\circ$ $\displaystyle 45^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times 50\pi}{\pi\times 20^2}=\frac{18000\pi}{200\pi}=45$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 50\pi}{\pi\times 20^2}=\frac{18000\pi}{200\pi}=45$

The central angle is $\displaystyle 45$ degrees.

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Example Question #113 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 12$ has an area of $48\pi$ $\displaystyle 48\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$110^\circ$ $\displaystyle 110^\circ$

$145^\circ$ $\displaystyle 145^\circ$

$130^\circ$ $\displaystyle 130^\circ$

$120^\circ$ $\displaystyle 120^\circ$

Correct answer:

$120^\circ$ $\displaystyle 120^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times 48\pi}{\pi\times 12^2}=\frac{17280\pi}{144\pi}=120$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 48\pi}{\pi\times 12^2}=\frac{17280\pi}{144\pi}=120$

The central angle is 120 $\displaystyle 120$ degrees.

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Example Question #114 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 3$ has an area of $\frac{1}{2}\pi$ $\displaystyle \frac{1}{2}\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$10^\circ$ $\displaystyle 10^\circ$

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

$80^\circ$ $\displaystyle 80^\circ$

$20^\circ$ $\displaystyle 20^\circ$

Correct answer:

$20^\circ$ $\displaystyle 20^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times \frac{1}{2}\pi}{\pi\times 3^2}=\frac{180\pi}{9\pi}=20$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times \frac{1}{2}\pi}{\pi\times 3^2}=\frac{180\pi}{9\pi}=20$

The central angle is $\displaystyle 20$ degrees.

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Example Question #115 : 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:

$92.5^\circ$ $\displaystyle 92.5^\circ$

$75.5^\circ$ $\displaystyle 75.5^\circ$

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

$132.5^\circ$ $\displaystyle 132.5^\circ$

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 #116 : Plane Geometry

A sector of a circle with a radius of $\displaystyle 5$ has an area of $3\pi$ $\displaystyle 3\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$43.2^\circ$ $\displaystyle 43.2^\circ$

$47.2^\circ$ $\displaystyle 47.2^\circ$

$45.2^\circ$ $\displaystyle 45.2^\circ$

$49.2^\circ$ $\displaystyle 49.2^\circ$

Correct answer:

$43.2^\circ$ $\displaystyle 43.2^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times 3\pi}{\pi\times 5^2}=\frac{1080\pi}{25\pi}=43.2$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 3\pi}{\pi\times 5^2}=\frac{1080\pi}{25\pi}=43.2$

The central angle is 43.2 $\displaystyle 43.2$ degrees.

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Example Question #117 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 4$ has an area of $\frac{1}{4}\pi$ $\displaystyle \frac{1}{4}\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$5.625^\circ$ $\displaystyle 5.625^\circ$

$6.225^\circ$ $\displaystyle 6.225^\circ$

$2.775^\circ$ $\displaystyle 2.775^\circ$

$8.555^\circ$ $\displaystyle 8.555^\circ$

Correct answer:

$5.625^\circ$ $\displaystyle 5.625^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times \frac{1}{4}\pi}{\pi\times 4^2}=\frac{90\pi}{16\pi}=5.625$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times \frac{1}{4}\pi}{\pi\times 4^2}=\frac{90\pi}{16\pi}=5.625$

The central angle is 5.625 $\displaystyle 5.625$ degrees.

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Example Question #118 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 6$ has an area of $14\pi$ $\displaystyle 14\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$120^\circ$ $\displaystyle 120^\circ$

$130^\circ$ $\displaystyle 130^\circ$

$150^\circ$ $\displaystyle 150^\circ$

$140^\circ$ $\displaystyle 140^\circ$

Correct answer:

$140^\circ$ $\displaystyle 140^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times 14\pi}{\pi\times 6^2}=\frac{5040\pi}{36\pi}=140$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 14\pi}{\pi\times 6^2}=\frac{5040\pi}{36\pi}=140$

The central angle is 140 $\displaystyle 140$ degrees.

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Example Question #119 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 6$ has an area of $35\pi$ $\displaystyle 35\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

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

$350^\circ$ $\displaystyle 350^\circ$

$330^\circ$ $\displaystyle 330^\circ$

$340^\circ$ $\displaystyle 340^\circ$

Correct answer:

$350^\circ$ $\displaystyle 350^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times 35\pi}{\pi\times 6^2}=\frac{12600\pi}{36\pi}=350$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 35\pi}{\pi\times 6^2}=\frac{12600\pi}{36\pi}=350$

The central angle is 350 $\displaystyle 350$ degrees.

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Example Question #120 : Plane Geometry

A sector in a circle with a radius of $\displaystyle 6$ has and area of $34\pi$ $\displaystyle 34\pi$. In degrees, what is the measurement of the central angle of the sector?

Possible Answers:

$355^\circ$ $\displaystyle 355^\circ$

$340^\circ$ $\displaystyle 340^\circ$

$350^\circ$ $\displaystyle 350^\circ$

$345^\circ$ $\displaystyle 345^\circ$

Correct answer:

$340^\circ$ $\displaystyle 340^\circ$

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.

$\text{Measurement of Central Angle}=\frac{360\times 34\pi}{\pi\times 6^2}=\frac{12240\pi}{36\pi}=340$ $\displaystyle \text{Measurement of Central Angle}=\frac{360\times 34\pi}{\pi\times 6^2}=\frac{12240\pi}{36\pi}=340$

The central angle is 340 $\displaystyle 340$ degrees.

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #111 : Plane Geometry

Example Question #112 : Plane Geometry

Example Question #113 : Plane Geometry

Example Question #114 : Plane Geometry

Example Question #115 : Plane Geometry

Example Question #116 : Plane Geometry

Example Question #117 : Plane Geometry

Example Question #118 : Plane Geometry

Example Question #119 : Plane Geometry

Example Question #120 : Plane Geometry

All Intermediate Geometry Resources

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