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

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All Common Core: High School - Geometry Resources

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Example Questions

Example Question #21 : Arc Length, Radii, Radian, And Sector Similarity And Proportionality: Ccss.Math.Content.Hsg C.B.5

Determine the area of the sector of a circle that has a central angle of degrees and an area of 100 .

Refer to the following figure to help calculate the solution.

Circlearc9 10

Possible Answers:

$\text{sector area}=25.80$

$\text{sector area}=25.8\overline{4}$

$\text{sector area}=25.5\overline{3}$

$\text{sector area}=25.8\overline{3}$

$\text{sector area}=25.3\overline{8}$

Correct answer:

$\text{sector area}=25.8\overline{3}$

Explanation:

State the known information,

$\\\text{Circle Area}= 100 \\\text{Central Angle}=93^\circ$

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,

$\\\frac{\theta}{360^\circ}=\frac{\text{sector area}}{100} \\\\\frac{93^\circ}{360^\circ}=\frac{\text{sector area}}{100} \\\\\frac{31}{120}=\frac{\text{sector area}}{100} \\\\\frac{31}{120}\times 100=\text{sector area} \\\\\text{sector area}=25.8\overline{3}$

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Example Question #71 : Circles

Calculate the arc length of a circle that has a central angle of degrees and an of 100 .

Refer to the following figure to help calculate the solution.

Circlearc9 10

Possible Answers:

s=9.185

s=9.851

s=9.158

s=9.518

s=9.150

Correct answer:

s=9.158

Explanation:

To calculate the arc length of a circle that has a central angle of degrees and a circumference of 100 , refer to the figure and the algebraic formula for arc length.

Circlearc9 10

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,

$\\s=\text{Arc Length} \\A=100 \\\theta=93^\circ\\A=\pi r^2\\C=2\pi r$

First, calculate the radius and the circumference.

$\\100=\pi r^2 \\\\\frac{100}{\pi}=r^2 \\\\\sqrt{\frac{100}{\pi}}=r$

$C=2\pi\sqrt{\frac{100}{\pi}}$

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

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

$s=\frac{93^\circ}{360^\circ}\times 2\pi\sqrt{\frac{100}{\pi}}$

s=9.158

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Example Question #23 : Arc Length, Radii, Radian, And Sector Similarity And Proportionality: Ccss.Math.Content.Hsg C.B.5

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.

Circlearc11 12

Possible Answers:

s=2.212

s=2.122

s=2.207

s=2.218

s=2.221

Correct answer:

s=2.212

Explanation:

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.

Circlearc11 12

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,

$\\s=\text{Arc Length} \\A=15 \\\theta=58^\circ\\A=\pi r^2\\C=2\pi r$

First, calculate the radius and the circumference.

$\\15=\pi r^2 \\\\\frac{15}{\pi}=r^2 \\\\\sqrt{\frac{15}{\pi}}=r$

$C=2\pi\sqrt{\frac{15}{\pi}}$

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

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

$s=\frac{58^\circ}{360^\circ}\times 2\pi\sqrt{\frac{15}{\pi}}$

s=2.212

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Example Question #72 : Circles

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.

Circlearc11 12

Possible Answers:

$\text{sector area}= 2.41\overline{6}$

$\text{sector area}= 2.61\overline{4}$

$\text{sector area}= 2.14\overline{6}$

$\text{sector area}= 2.46\overline{1}$

$\text{sector area}= 2.41\overline{9}$

Correct answer:

$\text{sector area}= 2.41\overline{6}$

Explanation:

State the known information,

$\\\text{Circle Area}= 15 \\\text{Central Angle}=58^\circ$

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,

$\\\frac{\theta}{360^\circ}=\frac{\text{sector area}}{15} \\\\\frac{58^\circ}{360^\circ}=\frac{\text{sector area}}{15} \\\\\frac{29}{180}=\frac{\text{sector area}}{15} \\\\\frac{29}{180}\times 15=\text{sector area} \\\\\text{sector area}=2.41\overline{6}$

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All Common Core: High School - Geometry Resources

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

Study concepts, example questions & explanations for Common Core: High School - Geometry

All Common Core: High School - Geometry Resources

Example Questions

Example Question #21 : Arc Length, Radii, Radian, And Sector Similarity And Proportionality: Ccss.Math.Content.Hsg C.B.5

Example Question #71 : Circles

Example Question #23 : Arc Length, Radii, Radian, And Sector Similarity And Proportionality: Ccss.Math.Content.Hsg C.B.5

Example Question #72 : Circles

All Common Core: High School - Geometry Resources

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