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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #51 : Circles

Find the area of the sector if it has a central angle of 107 $\displaystyle 107$ degrees and a radius of $\displaystyle 21$.

Possible Answers:

411.78 $\displaystyle 411.78$

395.52 $\displaystyle 395.52$

403.20 $\displaystyle 403.20$

449.02 $\displaystyle 449.02$

Correct answer:

411.78 $\displaystyle 411.78$

Explanation:

The circle in question can be drawn as shown by the figure below:

Since the area of a sector is just a fractional part of the area of a circle, we can write the following equation to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$, where $\displaystyle r$ is the radius of the circle.

Plug in the given central angle and radius to find the area of the sector.

$\text{Area of Sector}=\frac{107}{360}\pi(21)^2=411.78$ $\displaystyle \text{Area of Sector}=\frac{107}{360}\pi(21)^2=411.78$

Make sure to round to two places after the decimal.

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Example Question #51 : Circles

Find the area of a sector if it has a central angle of $\displaystyle 13$ degrees and a radius of $\displaystyle 3$.

Possible Answers:

1.49 $\displaystyle 1.49$

1.02 $\displaystyle 1.02$

1.33 $\displaystyle 1.33$

1.54 $\displaystyle 1.54$

Correct answer:

1.02 $\displaystyle 1.02$

Explanation:

The circle in question can be drawn as shown by the figure below:

Since the area of a sector is just a fractional part of the area of a circle, we can write the following equation to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$, where $\displaystyle r$ is the radius of the circle.

Plug in the given central angle and radius to find the area of the sector.

$\text{Area of Sector}=\frac{13}{360}\pi(3)^2=1.02$ $\displaystyle \text{Area of Sector}=\frac{13}{360}\pi(3)^2=1.02$

Make sure to round to two places after the decimal.

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Example Question #52 : Circles

Find the area of a sector if it has a central angle of $\displaystyle 42$ degrees and a radius of $\displaystyle 3$.

Possible Answers:

3.30 $\displaystyle 3.30$

4.81 $\displaystyle 4.81$

2.71 $\displaystyle 2.71$

3.29 $\displaystyle 3.29$

Correct answer:

3.30 $\displaystyle 3.30$

Explanation:

The circle in question can be drawn as shown by the figure below:

Since the area of a sector is just a fractional part of the area of a circle, we can write the following equation to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$, where $\displaystyle r$ is the radius of the circle.

Plug in the given central angle and radius to find the area of the sector.

$\text{Area of Sector}=\frac{42}{360}\pi(3)^2=3.30$ $\displaystyle \text{Area of Sector}=\frac{42}{360}\pi(3)^2=3.30$

Make sure to round to two places after the decimal.

Report an Error

Example Question #51 : Sectors

Find the area of the sector if it has a central angle of 164 $\displaystyle 164$ degrees and a radius of $\displaystyle 8$.

Possible Answers:

102.44 $\displaystyle 102.44$

114.12 $\displaystyle 114.12$

105.69 $\displaystyle 105.69$

91.59 $\displaystyle 91.59$

Correct answer:

91.59 $\displaystyle 91.59$

Explanation:

The circle in question can be drawn as shown by the figure below:

Since the area of a sector is just a fractional part of the area of a circle, we can write the following equation to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$, where $\displaystyle r$ is the radius of the circle.

Plug in the given central angle and radius to find the area of the sector.

$\text{Area of Sector}=\frac{164}{360}\pi(8)^2=91.59$ $\displaystyle \text{Area of Sector}=\frac{164}{360}\pi(8)^2=91.59$

Make sure to round to two places after the decimal.

Report an Error

Example Question #53 : Circles

Find the area of a sector if it has a central angle of 134 $\displaystyle 134$ degrees and a radius of $\displaystyle 17$.

Possible Answers:

337.95 $\displaystyle 337.95$

310.21 $\displaystyle 310.21$

456.42 $\displaystyle 456.42$

394.89 $\displaystyle 394.89$

Correct answer:

337.95 $\displaystyle 337.95$

Explanation:

The circle in question can be drawn as shown by the figure below:

Since the area of a sector is just a fractional part of the area of a circle, we can write the following equation to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$, where $\displaystyle r$ is the radius of the circle.

Plug in the given central angle and radius to find the area of the sector.

$\text{Area of Sector}=\frac{134}{360}\pi(17)^2=337.95$ $\displaystyle \text{Area of Sector}=\frac{134}{360}\pi(17)^2=337.95$

Make sure to round to two places after the decimal.

Report an Error

Example Question #54 : Circles

Find the area of a sector that has a central angle of 300 $\displaystyle 300$ degrees and a radius of $\displaystyle 14$.

Possible Answers:

490.05 $\displaystyle 490.05$

513.13 $\displaystyle 513.13$

551.29 $\displaystyle 551.29$

601.48 $\displaystyle 601.48$

Correct answer:

513.13 $\displaystyle 513.13$

Explanation:

The circle in question can be drawn as shown by the figure below:

Since the area of a sector is just a fractional part of the area of a circle, we can write the following equation to find the area of a sector:

$\text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$ $\displaystyle \text{Area of Sector}=\frac{\text{Central angle}}{360}\pi r^2$, where $\displaystyle r$ is the radius of the circle.

Plug in the given central angle and radius to find the area of the sector.

$\text{Area of Sector}=\frac{300}{360}\pi(14)^2=513.13$ $\displaystyle \text{Area of Sector}=\frac{300}{360}\pi(14)^2=513.13$

Make sure to round to two places after the decimal.

Report an Error

Example Question #55 : Circles

Find the area of a sector if it has an arc length of $\displaystyle 3$ and a radius of $\displaystyle 1$.

Possible Answers:

$\frac{7}{2}$ $\displaystyle \frac{7}{2}$

$\frac{5}{2}$ $\displaystyle \frac{5}{2}$

$\frac{3}{2}$ $\displaystyle \frac{3}{2}$

The area of the sector cannot be determined.

Correct answer:

$\frac{3}{2}$ $\displaystyle \frac{3}{2}$

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$ $\displaystyle \text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$ $\displaystyle \text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{3(1)}{2}=\frac{3}{2}$ $\displaystyle \text{Area}=\frac{3(1)}{2}=\frac{3}{2}$

Report an Error

Example Question #52 : Sectors

Find the area of a sector if it has an arc length of $\displaystyle 14$ and a radius of $\displaystyle 8$.

Possible Answers:

$\frac{55}{2}\pi$ $\displaystyle \frac{55}{2}\pi$

$\displaystyle 56$

$\displaystyle 60$

$52\pi$ $\displaystyle 52\pi$

Correct answer:

$\displaystyle 56$

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$ $\displaystyle \text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$ $\displaystyle \text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{14(8)}{2}=56$ $\displaystyle \text{Area}=\frac{14(8)}{2}=56$

Report an Error

Example Question #53 : Circles

Find the area of a sector if it has an arc length of $2\pi$ $\displaystyle 2\pi$ and a radius of $\displaystyle 6$.

Possible Answers:

19.02 $\displaystyle 19.02$

17.61 $\displaystyle 17.61$

20.31 $\displaystyle 20.31$

18.85 $\displaystyle 18.85$

Correct answer:

18.85 $\displaystyle 18.85$

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$ $\displaystyle \text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$ $\displaystyle \text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{2\pi(6)}{2}=18.85$ $\displaystyle \text{Area}=\frac{2\pi(6)}{2}=18.85$

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

Report an Error

Example Question #31 : How To Find The Area Of A Sector

Find the area of a sector if it has an arc length of $9\pi$ $\displaystyle 9\pi$ and a radius of $\displaystyle 10$.

Possible Answers:

195.67 $\displaystyle 195.67$

141.37 $\displaystyle 141.37$

155.30 $\displaystyle 155.30$

139.95 $\displaystyle 139.95$

Correct answer:

141.37 $\displaystyle 141.37$

Explanation:

The length of the arc of the sector is just a fraction of the arc of the circumference. The area of the sector will be the same fraction of the area as the length of the arc is of the circumference.

We can then write the following equation to find the area of the sector:

$\text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$ $\displaystyle \text{Area}=\frac{\text{Arc Length}}{2\pi r}\pi r^2$

The equation can be simplified to the following:

$\text{Area}=\frac{\text{Arc Length}(r)}{2}$ $\displaystyle \text{Area}=\frac{\text{Arc Length}(r)}{2}$

Plug in the given arc length and radius to find the area of the sector.

$\text{Area}=\frac{9\pi(10)}{2}=141.37$ $\displaystyle \text{Area}=\frac{9\pi(10)}{2}=141.37$

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

Report an Error

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Intermediate Geometry : Plane Geometry

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #51 : Circles

Example Question #51 : Circles

Example Question #52 : Circles

Example Question #51 : Sectors

Example Question #53 : Circles

Example Question #54 : Circles

Example Question #55 : Circles

Example Question #52 : Sectors

Example Question #53 : Circles

Example Question #31 : How To Find The Area Of A Sector

All Intermediate Geometry Resources

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