Find the Area of a Sector Using Radians - Pre-Calculus

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Question

Find the area of a sector if the radius of the circle is 4, and the angle of the sector is $\pi$ radians.

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Answer

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

Substitute the radius and theta.

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Question

Find the area of a sector if the radius is 1 and the angle of sector is $2\pi$ radians.

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Answer

Write the formula for the area of the sector in radians and substitute the given values.

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Question

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

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Answer

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

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

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

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Question

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

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Answer

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

Substitute the radius and the angle.

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Question

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

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Answer

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

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Question

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

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Answer

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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Question

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 , what is the area of the semi-circular design?

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Answer

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 , 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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Question

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 , what area will the solar panel cover?

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Answer

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 , what area will the solar panel cover?

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

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Question

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

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Answer

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:

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Question

Find the area of a sector if the radius of the circle is 4, and the angle of the sector is $\pi$ radians.

Tap to reveal answer

Answer

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

Substitute the radius and theta.

← Didn't Know|Knew It →

Question

Find the area of a sector if the radius is 1 and the angle of sector is $2\pi$ radians.

Tap to reveal answer

Answer

Write the formula for the area of the sector in radians and substitute the given values.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

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

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

Substitute the radius and the angle.

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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$

← Didn't Know|Knew It →

Question

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 , what is the area of the semi-circular design?

Tap to reveal answer

Answer

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 , 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$$

← Didn't Know|Knew It →

Question

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 , what area will the solar panel cover?

Tap to reveal answer

Answer

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 , what area will the solar panel cover?

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

← Didn't Know|Knew It →

Question

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

Tap to reveal answer

Answer

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:

← Didn't Know|Knew It →

Question

Find the area of a sector if the radius of the circle is 4, and the angle of the sector is $\pi$ radians.

Tap to reveal answer

Answer

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

Substitute the radius and theta.

← Didn't Know|Knew It →

Question

Find the area of a sector if the radius is 1 and the angle of sector is $2\pi$ radians.

Tap to reveal answer

Answer

Write the formula for the area of the sector in radians and substitute the given values.

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