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Precalculus : Angle Measures in Degrees and Radians

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Example Questions

Example Question #1 : Find The Area Of A Sector Using Radians

If the diameter of a circle is 6, find the area of a sector with a sector angle of $\pi$ .

Possible Answers:

$3\pi$

$6\pi$

$9\pi$

$18\pi$

$\frac{9\pi}{2}$

Correct answer:

$\frac{9\pi}{2}$

Explanation:

Write the formula for the area of the sector in radians.

$A=\frac{1}{2}r^2 \theta$

The given diameter is 6, which means the radius is 3.

Substitute both the radius and theta to solve for the area.

$A=\frac{1}{2}(3)^2 (\pi)= \frac{9\pi}{2}$

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Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector with the radius of 1 and angle of $\frac{\pi}{3}$ .

Possible Answers:

$\frac{\pi}{4}$

$\pi$

$\frac{\pi}{3}$

$\frac{\pi}{2}$

$\frac{\pi}{6}$

Correct answer:

$\frac{\pi}{6}$

Explanation:

Write the formula for the area of a sector in radians.

$A=\frac{1}{2}r^2\theta$

Substitute the radius and the angle.

$A=\frac{1}{2}(1)^2(\frac{\pi}{3})=\frac{\pi}{6}$

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Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector with a radius and angle of $\pi$ . Express the answer in terms of $\pi$ .

Possible Answers:

$\frac{\pi^3}{2}$

$\frac{\pi^4}{2}$

$\pi^3$

$\frac{\pi^2}{2}$

$\frac{\pi}{2}$

Correct answer:

$\frac{\pi^3}{2}$

Explanation:

Write the formula for the area of a sector. Substitute the radius and angle to solve for the area.

$A=\frac{1}{2}r^2 \theta= \frac{1}{2}\pi^2 \pi = \frac{\pi^3}{2}$

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Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector with a radius of 10 and an angle of $\frac{2}{5}\pi$ radians.

Possible Answers:

$4\pi$

$20\pi$

$4\pi^2$

$20\pi^2$

$40\pi$

Correct answer:

$20\pi^2$

Explanation:

Write the area of the sector in radians.

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

Substitute the givens and solve.

$A=\frac{1}{2}\pi (10)^2(\frac{2}{5}\pi) =20\pi ^2$

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Example Question #7 : Find The Area Of A Sector Using Radians

Katelyn is making a semi-circular design to put on one of her quilts. The design is not a perfect half-circle however, she needs to make the central angle $\frac{5\pi}{6}$ radians. If the radius of the circle is 15 cm , what is the area of the semi-circular design?

Possible Answers:

$93.75 \pi\: cm^2$

$93.75\: cm^2$

$937.5\: cm^2$

$45.75 \pi\: cm^2$

Correct answer:

$93.75 \pi\: cm^2$

Explanation:

Katelyn is making a semi-circular design to put on one of her quilts. The design is not a perfect half-circle however, she needs to make the central angle $\frac{5\pi}{6}$ radians. If the radius of the circle is $15 \:cm$ , what is the area of the semi-circular design?

Recall the following formual for area of a sector:

$A=\frac{Central\angle }{2\pi} *\pi r^2$

So, we plug in our knowns and solve for area!

$A=\frac{\frac{5\pi}{6} }{2\pi} *\pi {15}^2=\frac{5}{12}*225 \pi=93.75 \pi \:cm^2$

Therefore, our answer is...

$93.75 \pi\: cm^2$

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Example Question #8 : Find The Area Of A Sector Using Radians

Lucy is making a solar panel to cover a portion of a satellite dish. If the central angle of the sector the solar panel will cover is $\frac{4\pi}{3}radians$ , and the satellite dish has a radius of 1.03 m , what area will the solar panel cover?

Possible Answers:

0.71m^2

2.22m^2

22.2m^2

6.66m^2

Correct answer:

2.22m^2

Explanation:

To calculate area of a sector, use the following formula:

$AoS=\frac{\measuredangle Radians}{2\pi}\pi r^2$

Where the numerator of the fraction is the measure of the desired angle in radians, and r is the radius of the circle.

Now, we know both our variables, so we simply need to plug them in and simplify.

$AoS=\frac{\frac{4\pi}{3}}{2\pi}\pi (1.03)^2$

Now, this looks messy, but we can simplify it to get:

$AoS=\frac{2}{3}\pi (1.03)^2$

Next, use your calculator to find a decimal answer, and then round to get our final answer.

$AoS=\frac{2}{3}\pi (1.03)^2\approx 2.22$

Making the area of the sector

2.22m^2

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Example Question #1 : Find The Area Of A Sector Using Radians

Find the area of a sector in a circle, given that it encompasses 270^o of the actual circle, with a circle diameter of .

Possible Answers:

$S=\frac{39\pi}{4}$

$S=\frac{507\pi}{4}$

$S=\frac{39\pi}{2}$

$S=\frac{507\pi}{2}$

Correct answer:

$S=\frac{507\pi}{4}$

Explanation:

Equation for sector area is given by

$S=\frac{1}{2}\theta r^2$ , where $\theta$ is the angle measure of the sector in radians, and is the radius of the circle.

In our case, the sector encompasses $270^\circ$ of the circle or

$270deg\cdot\frac{\pi rad}{180deg}= \frac{3\pi}{2}rad$

To determine the radius , given diameter

$r=\frac{1}{2}d=\frac{26}{2}=13$

The sector area therefore is:

$S=\frac{1}{2}\cdot\frac{3\pi}{2}\cdot13^2= \frac{169\cdot3\pi}{4}= \frac{507\pi}{4}$

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Precalculus : Angle Measures in Degrees and Radians

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Area Of A Sector Using Radians

Example Question #1 : Find The Area Of A Sector Using Radians

Example Question #1 : Find The Area Of A Sector Using Radians

Example Question #1 : Find The Area Of A Sector Using Radians

Example Question #7 : Find The Area Of A Sector Using Radians

Example Question #8 : Find The Area Of A Sector Using Radians

Example Question #1 : Find The Area Of A Sector Using Radians

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