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Intermediate Geometry : How to find the area of a sector

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Example Questions

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.

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Example Question #61 : Circles

Find the area of a sector if it has an arc length of $3\pi$ $\displaystyle 3\pi$ and a radius of 5.7 $\displaystyle 5.7$.

Possible Answers:

26.86 $\displaystyle 26.86$

23.11 $\displaystyle 23.11$

25.99 $\displaystyle 25.99$

24.09 $\displaystyle 24.09$

Correct answer:

26.86 $\displaystyle 26.86$

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.

$\text{Area}=\frac{3\pi(5.7)}{2}=26.86$ $\displaystyle \text{Area}=\frac{3\pi(5.7)}{2}=26.86$

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

Report an Error

Example Question #61 : Plane Geometry

Find the area of a sector if it has an arc length of $10\pi$ $\displaystyle 10\pi$ and a radius of $\displaystyle 12$.

Possible Answers:

188.50 $\displaystyle 188.50$

171.41 $\displaystyle 171.41$

186.21 $\displaystyle 186.21$

192.59 $\displaystyle 192.59$

Correct answer:

188.50 $\displaystyle 188.50$

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.

$\text{Area}=\frac{12\pi(10)}{2}=188.50$ $\displaystyle \text{Area}=\frac{12\pi(10)}{2}=188.50$

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

Report an Error

Example Question #61 : Intermediate Geometry

Find the area of a sector if it has an arc length of $15\pi$ $\displaystyle 15\pi$ and a radius of 10.4 $\displaystyle 10.4$.

Possible Answers:

245.04 $\displaystyle 245.04$

214.81 $\displaystyle 214.81$

251.59 $\displaystyle 251.59$

199.44 $\displaystyle 199.44$

Correct answer:

245.04 $\displaystyle 245.04$

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.

$\text{Area}=\frac{15\pi(10.4)}{2}=245.04$ $\displaystyle \text{Area}=\frac{15\pi(10.4)}{2}=245.04$

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 $19\pi$ $\displaystyle 19\pi$ and a radius of $\displaystyle 50$.

Possible Answers:

1428.58 $\displaystyle 1428.58$

1521.20 $\displaystyle 1521.20$

1492.26 $\displaystyle 1492.26$

1304.51 $\displaystyle 1304.51$

Correct answer:

1492.26 $\displaystyle 1492.26$

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.

$\text{Area}=\frac{19\pi(50)}{2}=1492.26$ $\displaystyle \text{Area}=\frac{19\pi(50)}{2}=1492.26$

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

Report an Error

Example Question #61 : Plane Geometry

Find the area of a sector if it has an arc length of $\displaystyle 4$ and a radius of $12\pi$ $\displaystyle 12\pi$.

Possible Answers:

75.40 $\displaystyle 75.40$

81.08 $\displaystyle 81.08$

77.91 $\displaystyle 77.91$

90.36 $\displaystyle 90.36$

Correct answer:

75.40 $\displaystyle 75.40$

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.

$\text{Area}=\frac{4(12\pi)}{2}=75.40$ $\displaystyle \text{Area}=\frac{4(12\pi)}{2}=75.40$

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 $\displaystyle 15$ and a radius of $20\sqrt3$ $\displaystyle 20\sqrt3$.

Possible Answers:

241.20 $\displaystyle 241.20$

233.90 $\displaystyle 233.90$

215.29 $\displaystyle 215.29$

259.81 $\displaystyle 259.81$

Correct answer:

259.81 $\displaystyle 259.81$

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.

$\text{Area}=\frac{15(20\sqrt3)}{2}=259.81$ $\displaystyle \text{Area}=\frac{15(20\sqrt3)}{2}=259.81$

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

Report an Error

Example Question #67 : Circles

Find the area of a sector if it has an arc length of $13\pi$ $\displaystyle 13\pi$ and a radius of $49\sqrt3$ $\displaystyle 49\sqrt3$.

Possible Answers:

1441.20 $\displaystyle 1441.20$

1733.09 $\displaystyle 1733.09$

1720.91 $\displaystyle 1720.91$

1695.48 $\displaystyle 1695.48$

Correct answer:

1733.09 $\displaystyle 1733.09$

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.

$\text{Area}=\frac{13\pi(49\sqrt3)}{2}=1733.09$ $\displaystyle \text{Area}=\frac{13\pi(49\sqrt3)}{2}=1733.09$

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

Report an Error

Example Question #32 : How To Find The Area Of A Sector

Find the area of a sector if it has an arc length of $\displaystyle 19$ and a radius of $20\sqrt2$ $\displaystyle 20\sqrt2$.

Possible Answers:

268.70 $\displaystyle 268.70$

255.64 $\displaystyle 255.64$

210.40 $\displaystyle 210.40$

249.17 $\displaystyle 249.17$

Correct answer:

268.70 $\displaystyle 268.70$

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.

$\text{Area}=\frac{19(20\sqrt2)}{2}=268.70$ $\displaystyle \text{Area}=\frac{19(20\sqrt2)}{2}=268.70$

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

Report an Error

Example Question #33 : How To Find The Area Of A Sector

Find the area of a sector if it has a central angle of 172 $\displaystyle 172$ degrees and a radius of $\displaystyle 16$.

Possible Answers:

384.25 $\displaystyle 384.25$

891.41 $\displaystyle 891.41$

701.02 $\displaystyle 701.02$

722.39 $\displaystyle 722.39$

Correct answer:

384.25 $\displaystyle 384.25$

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.

$\text{Area of Sector}=\frac{172}{360}\pi(16)^2=384.25$ $\displaystyle \text{Area of Sector}=\frac{172}{360}\pi(16)^2=384.25$

Make sure to round to two places after the decimal.

Report an Error

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Intermediate Geometry : How to find the area of a sector

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

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

Example Question #61 : Circles

Example Question #61 : Plane Geometry

Example Question #61 : Intermediate Geometry

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

Example Question #61 : Plane Geometry

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

Example Question #67 : Circles

Example Question #32 : How To Find The Area Of A Sector

Example Question #33 : How To Find The Area Of A Sector

All Intermediate Geometry Resources

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