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Area of Sectors and Segments (Radians)Area of Sectors and Segments (Radians)
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Question 1 of 4
1. Question
Find the area of the shaded region
Use `pi=3.1415`
Round your answer to two decimal places- `A=` (23.56) `\text(cm)^2`
Hint
Help VideoCorrect
Well Done!
Incorrect
Help VideoArea of a Sector
$$A=\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$$Converting Degrees to Radian
`\text(radian)=\text(degrees)xxpi/(180°)`First, convert the degree into radian`\text(radian)` `=` `\text(degrees)xxpi/(180°)` `=` `75°xxpi/(180°)` `=` `(75°pi)/(180°)` `=` `(5pi)/12` Simplify Next, substitute the known values and solve for angle of the shaded region`r` `=` `6` `theta` `=` `(5pi)/12` `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 `pi=3.1415` `=` `18*1.3089` `=` `23.56` Rounded to two decimal places `A=23.56 \text(cm)^2` -
Question 2 of 4
2. Question
Find the area of the shaded region
Use `pi=3.14159`
Round your answer to two decimal places- `A=` (1837.83) `\text(cm)^2`
Hint
Help VideoCorrect
Nice Job!
Incorrect
Help VideoArea of a Sector
$$A=\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$$Converting Degrees to Radian
`\text(radian)=\text(degrees)xxpi/(180°)`First, convert the degree into radian`\text(radian)` `=` `\text(degrees)xxpi/(180°)` `=` `150°xxpi/(180°)` `=` `(150°pi)/(180°)` `=` `(5pi)/6` Simplify Next, find the radius of the smaller sector by subtracting the radius of the larger sector to the total radius`\text(radius)(\text(Smaller Sector))` `=` `40-26` `=` `14` Then, substitute the known values and solve for angle of the whole sector`r` `=` `40` `theta` `=` `(5pi)/12` `\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` Rounded to two decimal places Next, substitute the known values and solve for angle of the smaller sector`r` `=` `14` `theta` `=` `(5pi)/12` `\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 `=` `81.667xxpi` `=` `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`\text(Area)(\text(Shaded Region))` `=` `\text(Area)(\text(Whole Sector))``-``\text(Area)(\text(Smaller Sector))` `=` `2094.39``-``256.56` `=` `1837.83` `A=1837.83 \text(cm)^2` -
Question 3 of 4
3. Question
Find the area of the purple shaded region
Hint
Help VideoCorrect
Excellent!
Incorrect
Help VideoArea of a Sector
$$A=\frac{1}{2}\color{#9a00c7}{r}^2\color{#e65021}{\theta}$$Converting Degrees to Radian
`\text(radian)=\text(degrees)xxpi/(180°)`First, convert the degree into radian`\text(radian)` `=` `\text(degrees)xxpi/(180°)` `=` `60°xxpi/(180°)` `=` `(60°pi)/(180°)` `=` `(pi)/3` Simplify Next, substitute the known values and solve for angle of the whole sector`r` `=` `70` `theta` `=` `(pi)/3` `\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` `=` `50` `theta` `=` `(pi)/3` `\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`\text(Area)(\text(Shaded Region))` `=` `\text(Area)(\text(Whole Sector))``-``\text(Area)(\text(Smaller Sector))` `=` `(2450pi)/3``-``(1250pi)/3` `=` `(1200pi)/3` `=` `400pi` Simplify `A=400pi \text(cm)^2` -
Question 4 of 4
4. Question
Find the area of the shaded region
Use `pi=3.14`
Round your answer to three decimal places- `A=` (5.637) `\text(cm)^2`
Hint
Help VideoCorrect
Fantastic!
Incorrect
Help VideoArea of a Segment
$$A=\frac{1}{2}\color{#9a00c7}{r}^2(\color{#e65021}{\theta}-\text{sin}\color{#e65021}{\theta})$$Converting Degrees to Radian
`\text(radian)=\text(degrees)xxpi/(180°)`First, convert the degree into radian`\text(radian)` `=` `\text(degrees)xxpi/(180°)` `=` `45°xxpi/(180°)` `=` `(45°pi)/(180°)` `=` `(pi)/4` Simplify Next, substitute the known values and solve for angle of the shaded region`r` `=` `12` `theta` `=` `(pi)/4` `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*(0.785-0.7071)` Use `pi=3.14` `=` `72*0.0783` `=` `5.637` Rounded to three decimal places `A=5.637 \text(cm)^2`