Arc Length, Radii, Radian, and Sector Similarity and Proportionality: CCSS.Math.Content.HSG-C.B.5 - Geometry

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Question

If the radius is , and the central angle is $\frac{\pi}{5}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 13 \cdot \frac{\pi}{5} \s = \frac{13 \pi}{5}$

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Question

If the radius is , and the central angle is $\frac{\pi}{23}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 10 \cdot \frac{\pi}{23} \s = \frac{10 \pi}{23}$

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Question

If the radius is , and the central angle is $\frac{\pi}{23}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 15 \cdot \frac{\pi}{23} \s = \frac{15 \pi}{23}$

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Question

If the radius is , and the central angle is $\frac{\pi}{33}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 12 \cdot \frac{\pi}{33} \s = \frac{4 \pi}{11}$

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Question

If the radius is , and the central angle is $\frac{\pi}{24}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 6 \cdot \frac{\pi}{24} \s = \frac{\pi}{4}$

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Question

If the radius is , and the central angle is $\frac{\pi}{31}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 19 \cdot \frac{\pi}{31} \s = \frac{19 \pi}{31}$

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Question

If the radius is , and the central angle is $\frac{\pi}{26}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 8 \cdot \frac{\pi}{26} \s = \frac{4 \pi}{13}$

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Question

If the radius is , and the central angle is $\frac{\pi}{4}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 19 \cdot \frac{\pi}{4} \s = \frac{19 \pi}{4}$

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Question

If the radius is , and the central angle is $\frac{\pi}{37}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 5 \cdot \frac{\pi}{37} \s = \frac{5 \pi}{37}$

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Question

If the radius is , and the central angle is $\frac{\pi}{24}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 5 \cdot \frac{\pi}{24} \s = \frac{5 \pi}{24}$

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Question

If the radius is , and the central angle is $\frac{\pi}{14}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 13 \cdot \frac{\pi}{14} \s = \frac{13 \pi}{14}$

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Question

If the radius is , and the central angle is $\frac{\pi}{20}$ find the arc length.

Answer

To find the arc length, we simply multiply the radius by the central angle.

$\s = \theta r \s = 16 \cdot \frac{\pi}{20} \s = \frac{4 \pi}{5}$

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Question

Calculate the arc length of a circle that has a central angle of degrees and a circumference of .

Refer to the following figure to help calculate the solution.

Answer

To calculate the arc length of a circle that has a central angle of degrees and a circumference of , refer to the figure and the algebraic formula for arc length.

The algebraic formula for arc length is as follows.

$s=\frac{\theta}{360^\circ}\times C$

where,

$\s=\text{Arc Length} \C=\text{Circumference} \\theta=\text{Central Angle}$

For this particular question the known information is,

$\s=\text{Arc Length} \C=14 \\theta=33^\circ$

Substitute these values into the formula and solve for the arc length.

$s=\frac{\theta}{360^\circ}\times C$

$s=\frac{33^\circ}{360^\circ}\times 14$

$s=1.28\overline{3}$

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Question

Determine the area of the sector of a circle that has a central angle of degrees and a circumference of .

Refer to the following figure to help calculate the solution.

Answer

To calculate the area of the sector of the circle all that is needed is the central angle and the area of the circle. Since the central angle and circumference are given the area of the circle can be calculated.

The formulas to find the area and circumference of a circle are as follows.

$\A=\pi r^2 \C=2\pi r$

The known information is,

$\\theta =33^\circ \C=14$

Solving for the radius,

$\14=2\pi r \7=\pi r \\r=\frac{7}{\pi}$

Now the area of the whole circle is,

$\A=\pi r^2 \A=\pi \left(\frac{7}{\pi} \right )^2 \\A=\pi \times \frac{49}{\pi^2} \\A=\frac{49}{\pi}$

Now, since the question is asking for the area of the sector, a ratio will need to be constructed. Recall that a circle is composed of 360 degrees. Therefore, the following ratio can be made,

$\\frac{\theta}{360^\circ}=\frac{\text{sector area}}{\frac{49}{\pi}} \\\frac{33^\circ}{360^\circ}=\frac{\text{sector area}}{\frac{49}{\pi}} \\\frac{11}{120}=\frac{\text{sector area}}{\frac{49}{\pi}} \\\frac{11}{120}\times \frac{49}{\pi}=\text{sector area} \\\text{sector area}=1.43$

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Question

Calculate the arc length of a circle that has a central angle of degrees and a circumference of .

Refer to the following figure to help calculate the solution.

Answer

To calculate the arc length of a circle that has a central angle of degrees and a circumference of , refer to the figure and the algebraic formula for arc length.

The algebraic formula for arc length is as follows.

$s=\frac{\theta}{360^\circ}\times C$

where,

$\s=\text{Arc Length} \C=\text{Circumference} \\theta=\text{Central Angle}$

For this particular question the known information is,

$\s=\text{Arc Length} \C=20 \\theta=105^\circ$

Substitute these values into the formula and solve for the arc length.

$s=\frac{\theta}{360^\circ}\times C$

$s=\frac{105^\circ}{360^\circ}\times 20$

$s=5.8\overline{3}$

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Question

Determine the area of the sector of a circle that has a central angle of degrees and a circumference of .

Refer to the following figure to help calculate the solution.

Answer

To calculate the area of the sector of the circle all that is needed is the central angle and the area of the circle. Since the central angle and circumference are given the area of the circle can be calculated.

The formulas to find the area and circumference of a circle are as follows.

$\A=\pi r^2 \C=2\pi r$

The known information is,

$\\theta =105^\circ \C=20$

Solving for the radius,

$\20=2\pi r \10=\pi r \\r=\frac{10}{\pi}$

Now the area of the whole circle is,

$\A=\pi r^2 \A=\pi \left(\frac{10}{\pi} \right )^2 \\A=\pi \times \frac{100}{\pi^2} \\A=\frac{100}{\pi}$

Now, since the question is asking for the area of the sector, a ratio will need to be constructed. Recall that a circle is composed of 360 degrees. Therefore, the following ratio can be made,

$\\frac{\theta}{360^\circ}=\frac{\text{sector area}}{\frac{100}{\pi}} \\\frac{105^\circ}{360^\circ}=\frac{\text{sector area}}{\frac{100}{\pi}} \\\frac{7}{24}=\frac{\text{sector area}}{\frac{100}{\pi}} \\\frac{7}{24}\times \frac{100}{\pi}=\text{sector area} \\\text{sector area}=9.28$

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Question

Calculate the arc length of a circle that has a central angle of degrees and a circumference of .

Refer to the following figure to help calculate the solution.

Answer

To calculate the arc length of a circle that has a central angle of degrees and a circumference of , refer to the figure and the algebraic formula for arc length.

The algebraic formula for arc length is as follows.

$s=\frac{\theta}{360^\circ}\times C$

where,

$\s=\text{Arc Length} \C=\text{Circumference} \\theta=\text{Central Angle}$

For this particular question the known information is,

$\s=\text{Arc Length} \C=16 \\theta=123^\circ$

Substitute these values into the formula and solve for the arc length.

$s=\frac{\theta}{360^\circ}\times C$

$s=\frac{123^\circ}{360^\circ}\times 16$

$s=5.4\overline{6}$

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Question

Determine the area of the sector of a circle that has a central angle of degrees and a circumference of .

Refer to the following figure to help calculate the solution.

Answer

To calculate the area of the sector of the circle all that is needed is the central angle and the area of the circle. Since the central angle and circumference are given the area of the circle can be calculated.

The formulas to find the area and circumference of a circle are as follows.

$\A=\pi r^2 \C=2\pi r$

The known information is,

$\\theta =123^\circ \C=16$

Solving for the radius,

$\16=2\pi r \8=\pi r \\r=\frac{8}{\pi}$

Now the area of the whole circle is,

$\A=\pi r^2 \A=\pi \left(\frac{8}{\pi} \right )^2 \\A=\pi \times \frac{64}{\pi^2} \\A=\frac{64}{\pi}$

Now, since the question is asking for the area of the sector, a ratio will need to be constructed. Recall that a circle is composed of 360 degrees. Therefore, the following ratio can be made,

$\\frac{\theta}{360^\circ}=\frac{\text{sector area}}{\frac{64}{\pi}} \\\frac{123^\circ}{360^\circ}=\frac{\text{sector area}}{\frac{64}{\pi}} \\\frac{41}{120}=\frac{\text{sector area}}{\frac{64}{\pi}} \\\frac{41}{120}\times \frac{64}{\pi}=\text{sector area} \\\text{sector area}=6.96$

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Question

Calculate the arc length of a circle that has a central angle of degrees and an of .

Refer to the following figure to help calculate the solution.

Answer

To calculate the arc length of a circle that has a central angle of degrees and a circumference of , refer to the figure and the algebraic formula for arc length.

The algebraic formula for arc length is as follows.

$s=\frac{\theta}{360^\circ}\times C$

where,

$\s=\text{Arc Length} \C=\text{Circumference} \\theta=\text{Central Angle}\A=Area$

For this particular question the known information is,

$\s=\text{Arc Length} \A=46 \\theta=26^\circ\A=\pi r^2\C=2\pi r$

First, calculate the radius and the circumference.

$\46=\pi r^2 \\\frac{46}{\pi}=r^2 \\\sqrt{\frac{46}{\pi}}=r$

$C=2\pi\sqrt{\frac{46}{\pi}}$

Substitute these values into the formula and solve for the arc length.

$s=\frac{\theta}{360^\circ}\times C$

$s=\frac{26^\circ}{360^\circ}\times 2\pi\sqrt{\frac{46}{\pi}}$

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Question

Determine the area of the sector of a circle that has a central angle of degrees and an area of .

Refer to the following figure to help calculate the solution.

Answer

State the known information,

$\\text{Circle Area}= 46 \\text{Central Angle}=26^\circ$

Since the question is asking for the area of the sector, a ratio will need to be constructed. Recall that a circle is composed of 360 degrees. Therefore, the following ratio can be made,

$\\frac{\theta}{360^\circ}=\frac{\text{sector area}}{46} \\\frac{26^\circ}{360^\circ}=\frac{\text{sector area}}{46} \\\frac{13}{180}=\frac{\text{sector area}}{46} \\\frac{13}{180}\times 46=\text{sector area} \\\text{sector area}=3.\overline{22}$

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