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

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

Substitute the radius and theta.

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

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

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

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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.

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

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

Substitute the radius and the angle.

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

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

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

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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$

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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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$$

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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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

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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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:

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 see back →

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

Substitute the radius and theta.

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

Tap to see back →

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

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

Tap to see back →

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.

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

Tap to see back →

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

Substitute the radius and the angle.

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

Tap to see back →

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

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

Tap to see back →

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$

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 see back →

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$$

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 see back →

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

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

Tap to see back →

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:

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 see back →

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

Substitute the radius and theta.

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

Tap to see back →

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