Topics
>
Trigonometry>
Radians, Arc Length and Area>
Areas of Sectors and Segments (Radians)>
Areas of Sectors and Segments (Radians)Areas of Sectors and Segments (Radians)
Try VividMath Premium to unlock full access
Time limit: 0
Quiz summary
0 of 4 questions completed
Questions:
- 1
- 2
- 3
- 4
Information
–
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading...
You must sign in or sign up to start the quiz.
You have to finish following quiz, to start this quiz:
Loading...
- 1
- 2
- 3
- 4
- Answered
- Review
-
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
Area 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
Area 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
Area 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
Area 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`