Areas of Sectors and Segments (Radians)

Question 1 of 4

Find the area of the shaded region

Use

π = 3.1415

$pi=3.1415$
Round your answer to two decimal places

$A =$ $A=$ (23.56) ${cm}^{2}$ $\text(cm)^2$

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Area of a Sector

A = \frac{1}{2} r^{2} θ

$A=\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$

Converting Degrees to Radian

radian = degrees \times \frac{π}{180 °}

$\text(radian)=\text(degrees)xxpi/(180°)$

First, convert the degree into radian

$radian$ $\text(radian)$	$=$ $=$	$degrees \times \frac{π}{180 °}$ $\text(degrees)xxpi/(180°)$

	$=$ $=$	$75 ° \times \frac{π}{180 °}$ $75°xxpi/(180°)$

	$=$ $=$	$\frac{75 ° π}{180 °}$ $(75°pi)/(180°)$

	$=$ $=$	$\frac{5 π}{12}$ $(5pi)/12$	Simplify

Next, substitute the known values and solve for angle of the shaded region

$r$ $r$	$=$ $=$	$6$ $6$

$θ$ $theta$	$=$ $=$	$\frac{5 π}{12}$ $(5pi)/12$

$A$ $A$	$=$ $=$	$\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$

	$=$ $=$	$\frac{1}{2}\cdot\color{#9a00c7}{6}^2\cdot\color{#e65021}{\frac{5\pi}{12}}$	Substitute known values

	$=$ $=$	$\frac{1}{2}\cdot36\cdot{\frac{15.7075}{12}}$	Use $π = 3.1415$ $pi=3.1415$

	$=$ $=$	$18 \cdot 1.3089$ $18*1.3089$
	$=$ $=$	$23.56$ $23.56$	Rounded to two decimal places

A = 23.56 {cm}^{2}

$A=23.56 \text(cm)^2$

Question 2 of 4

Find the area of the shaded region

Use

π = 3.14159

$pi=3.14159$
Round your answer to two decimal places

$A =$ $A=$ (1837.83) ${cm}^{2}$ $\text(cm)^2$

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Mute

Area of a Sector

A = \frac{1}{2} r^{2} θ

$A=\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$

Converting Degrees to Radian

radian = degrees \times \frac{π}{180 °}

$\text(radian)=\text(degrees)xxpi/(180°)$

First, convert the degree into radian

$radian$ $\text(radian)$	$=$ $=$	$degrees \times \frac{π}{180 °}$ $\text(degrees)xxpi/(180°)$

	$=$ $=$	$150 ° \times \frac{π}{180 °}$ $150°xxpi/(180°)$

	$=$ $=$	$\frac{150 ° π}{180 °}$ $(150°pi)/(180°)$

	$=$ $=$	$\frac{5 π}{6}$ $(5pi)/6$	Simplify

Next, find the radius of the smaller sector by subtracting the radius of the larger sector to the total radius

$radius (Smaller sector)$ $\text(radius)(\text(Smaller Sector))$	$=$ $=$	$40 - 26$ $40-26$
	$=$ $=$	$14$ $14$

Then, substitute the known values and solve for angle of the whole sector

$r$ $r$	$=$ $=$	$40$ $40$

$θ$ $theta$	$=$ $=$	$\frac{5 π}{12}$ $(5pi)/12$

$Area (Whole sector)$ $\text(Area)(\text(Whole Sector))$	$=$ $=$	$\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$

	$=$ $=$	$\frac{1}{2}\cdot\color{#9a00c7}{40}^2\cdot\color{#e65021}{\frac{5\pi}{6}}$	Substitute known values

	$=$ $=$	$\frac{1}{2}\cdot1600\cdot{\frac{5\pi}{6}}$	Evaluate

	$=$ $=$	$\frac{2000}{3}\cdot\pi$	Simplify

	$=$ $=$	$2094.39$ $2094.39$	Rounded to two decimal places

Next, substitute the known values and solve for angle of the smaller sector

$r$ $r$	$=$ $=$	$14$ $14$

$θ$ $theta$	$=$ $=$	$\frac{5 π}{12}$ $(5pi)/12$

$Area (Smaller sector)$ $\text(Area)(\text(Smaller Sector))$	$=$ $=$	$\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$

	$=$ $=$	$\frac{1}{2}\cdot\color{#9a00c7}{14}^2\cdot\color{#e65021}{\frac{5\pi}{6}}$	Substitute known values

	$=$ $=$	$\frac{1}{2}\cdot196\cdot{\frac{5\pi}{6}}$	Evaluate

	$=$ $=$	$\frac{98\cdot5}{6}\cdot\pi$	Simplify

	$=$ $=$	$81.667 \times π$ $81.667xxpi$

	$=$ $=$	$256.56$ $256.56$	Rounded to two decimal places

Finally, subtract the area of the smaller sector from the area of the whole sector to get the area of the shaded region

$Area (Shaded Region)$ $\text(Area)(\text(Shaded Region))$	$=$ $=$	$Area (Whole sector)$ $\text(Area)(\text(Whole Sector))$ $-$ $-$ $Area (Smaller sector)$ $\text(Area)(\text(Smaller Sector))$
	$=$ $=$	$2094.39$ $2094.39$ $-$ $-$ $256.56$ $256.56$
	$=$ $=$	$1837.83$ $1837.83$

A = 1837.83 {cm}^{2}

$A=1837.83 \text(cm)^2$

Question 3 of 4

Find the area of the purple shaded region

1. $A = 320 π {cm}^{2}$ $A=320pi \text(cm)^2$
2. $A = 450 π {cm}^{2}$ $A=450pi \text(cm)^2$
3. $A = 500 π {cm}^{2}$ $A=500pi \text(cm)^2$
4. $A = 400 π {cm}^{2}$ $A=400pi \text(cm)^2$

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Mute

Area of a Sector

A = \frac{1}{2} r^{2} θ

$A=\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$

Converting Degrees to Radian

radian = degrees \times \frac{π}{180 °}

$\text(radian)=\text(degrees)xxpi/(180°)$

First, convert the degree into radian

$radian$ $\text(radian)$	$=$ $=$	$degrees \times \frac{π}{180 °}$ $\text(degrees)xxpi/(180°)$

	$=$ $=$	$60 ° \times \frac{π}{180 °}$ $60°xxpi/(180°)$

	$=$ $=$	$\frac{60 ° π}{180 °}$ $(60°pi)/(180°)$

	$=$ $=$	$\frac{π}{3}$ $(pi)/3$	Simplify

Next, substitute the known values and solve for angle of the whole sector

$r$ $r$	$=$ $=$	$70$ $70$

$θ$ $theta$	$=$ $=$	$\frac{π}{3}$ $(pi)/3$

$Area (Whole sector)$ $\text(Area)(\text(Whole Sector))$	$=$ $=$	$\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$

	$=$ $=$	$\frac{1}{2}\cdot\color{#9a00c7}{70}^2\cdot\color{#e65021}{\frac{\pi}{3}}$	Substitute known values

	$=$ $=$	$\frac{2450\pi}{3}$	Simplify

Next, substitute the known values and solve for angle of the smaller sector

$r$ $r$	$=$ $=$	$50$ $50$

$θ$ $theta$	$=$ $=$	$\frac{π}{3}$ $(pi)/3$

$Area (Smaller sector)$ $\text(Area)(\text(Smaller Sector))$	$=$ $=$	$\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$

	$=$ $=$	$\frac{1}{2}\cdot\color{#9a00c7}{50}^2\cdot\color{#e65021}{\frac{\pi}{3}}$	Substitute known values

	$=$ $=$	$\frac{1250\pi}{3}$	Simplify

Finally, subtract the area of the smaller sector from the area of the whole sector to get the area of the shaded region

$Area (Shaded Region)$ $\text(Area)(\text(Shaded Region))$	$=$ $=$	$Area (Whole sector)$ $\text(Area)(\text(Whole Sector))$ $-$ $-$ $Area (Smaller sector)$ $\text(Area)(\text(Smaller Sector))$

	$=$ $=$	$\frac{2450 π}{3}$ $(2450pi)/3$ $-$ $-$ $\frac{1250 π}{3}$ $(1250pi)/3$

	$=$ $=$	$\frac{1200 π}{3}$ $(1200pi)/3$

	$=$ $=$	$400 π$ $400pi$	Simplify

A = 400 π {cm}^{2}

$A=400pi \text(cm)^2$

Question 4 of 4

Find the area of the shaded region

Use

π = 3.14

$pi=3.14$
Round your answer to three decimal places

$A =$ $A=$ (5.637) ${cm}^{2}$ $\text(cm)^2$

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Area of a Segment

A = \frac{1}{2} r^{2} (θ - sin θ)

$A=\frac{1}{2}\color{#9a00c7}{r}^2(\color{#e65021}{\theta}-\text{sin}\color{#e65021}{\theta})$

Converting Degrees to Radian

radian = degrees \times \frac{π}{180 °}

$\text(radian)=\text(degrees)xxpi/(180°)$

First, convert the degree into radian

$radian$ $\text(radian)$	$=$ $=$	$degrees \times \frac{π}{180 °}$ $\text(degrees)xxpi/(180°)$

	$=$ $=$	$45 ° \times \frac{π}{180 °}$ $45°xxpi/(180°)$

	$=$ $=$	$\frac{45 ° π}{180 °}$ $(45°pi)/(180°)$

	$=$ $=$	$\frac{π}{4}$ $(pi)/4$	Simplify

Next, substitute the known values and solve for angle of the shaded region

$r$ $r$	$=$ $=$	$12$ $12$

$θ$ $theta$	$=$ $=$	$\frac{π}{4}$ $(pi)/4$

$A$ $A$	$=$ $=$	$\frac{1}{2}\color{#9a00c7}{r}^2(\color{#e65021}{\theta}-\text{sin}\color{#e65021}{\theta})$

	$=$ $=$	$\frac{1}{2}\cdot\color{#9a00c7}{12}^2\cdot\left(\color{#e65021}{\frac{\pi}{4}}-\text{sin}\color{#e65021}{\frac{\pi}{4}}\right)$	Substitute known values

	$=$ $=$	$\frac{1}{2}\cdot144\cdot\left({\frac{\pi}{4}-\frac{1}{\sqrt{2}}}\right)$

	$=$ $=$	$72 \cdot (0.785 - 0.7071)$ $72*(0.785-0.7071)$	Use $π = 3.14$ $pi=3.14$
	$=$ $=$	$72 \cdot 0.0783$ $72*0.0783$
	$=$ $=$	$5.637$ $5.637$	Rounded to three decimal places

A = 5.637 {cm}^{2}

$A=5.637 \text(cm)^2$

Areas of Sectors and Segments (Radians)

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Quiz summary

Information

1. Question

Hint

Well Done!

Area of a Sector

Converting Degrees to Radian

2. Question

Hint

Nice Job!

Area of a Sector

Converting Degrees to Radian

3. Question

Hint

Excellent!

Area of a Sector

Converting Degrees to Radian

4. Question

Hint

Fantastic!

Area of a Segment

Converting Degrees to Radian

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