Lesson Objectives
- Learn how to find arc length on a circle
- Learn how to find the area of a sector of a circle
How to Find the Arc Length on a Circle
In our last lesson, we learned how to measure angles using radians. In general, if our angle θ is a central angle of a circle of radius r, and θ intercepts an arc of length s, then we can obtain our radian measure of θ as: $$θ=\frac{s}{r}$$ Therefore, if the arc length (s) is equal to the radius (r), the measure of the angle will be exactly 1 radian. 
$$s=r$$ $$θ=\frac{s}{r}$$ $$θ=\frac{r}{r}=1$$ Note: when working with angles, if the degree symbol "°" is not shown, our measure is given in terms of radians. 
We know that the measure of our angle θ in radians is represented by: $$θ=\frac{s}{r}$$ If we multiply both sides by r, we will obtain: $$\frac{s}{r}=θ$$ $$\require{cancel}\cancel{r}\cdot \frac{s}{\cancel{r}}=r \cdot θ$$ $$s=r θ$$ This leads us to the following formula:
Example #1: Find the length of each arc.
$$θ=\frac{π}{4}$$ $$r=12\hspace{.2em}\text{miles}$$ To find the length of our given arc, let's use our formula: $$s=r θ$$ Let's plug into our formula: $$s=12 \hspace{.2em}\text{miles}\cdot \frac{π}{4}$$ $$s=3\cancel{12}\hspace{.2em}\text{miles}\cdot \frac{π}{\cancel{4}}$$ $$s=3π \hspace{.2em}\text{miles}$$ Example #2: Find the length of each arc. 
$$θ=135°$$ $$r=12\hspace{.2em}\text{inches}$$ In this case, we can't immediately use our formula since our angle measure is given in degrees. We need to first convert θ to radians. $$135° \cdot \frac{π}{180°}=\frac{3π}{4}$$ Now that θ has been converted into radians, we can use our formula: $$s=r θ$$ $$s=12 \hspace{.2em}\text{inches}\cdot \frac{3π}{4}$$ $$s=3\cancel{12}\hspace{.2em}\text{inches}\cdot \frac{3π}{\cancel{4}}$$ $$s=9π \hspace{.2em}\text{inches}$$ 
The formula for the area of a complete circle with a radius r: $$\text{area of a circle}=π r^2$$ 
A sector of a circle is the portion of the interior of a circle intercepted by a central angle. Visually, it looks like a "piece of pie". 
If a central angle for a sector of a circle has measure θ radians, then the sector makes up the fraction: $\frac{θ}{2π}$ of a complete circle. To obtain the area of a sector, we will modify our area of a circle formula:
Example #3: Find the area of each sector.
$$θ=\frac{3π}{2}$$ $$r=9 \hspace{.1em}\text{meters}$$ Let's plug into our formula: $$\frac{1}{2}r^2 θ$$ $$\frac{1}{2}\cdot (9 \hspace{.1em}\text{meters})^2 \cdot \frac{3π}{2}$$ $$\frac{243 π}{4}\hspace{.1em}\text{square meters}$$ Example #4: Find the area of each sector. 
$$θ=60°$$ $$r=13 \hspace{.1em}\text{yards}$$ Since θ is given in degrees, our first step is to convert θ into radians. $$60° \cdot \frac{π}{180°}=\frac{π}{3}$$ Now, we can plug into our formula: $$\frac{1}{2}r^2 θ$$ $$\frac{1}{2}\cdot (13 \hspace{.1em}\text{yards})^2 \cdot \frac{π}{3}$$ $$\frac{169 π}{6}\hspace{.1em}\text{square yards}$$
$$s=r$$ $$θ=\frac{s}{r}$$ $$θ=\frac{r}{r}=1$$ Note: when working with angles, if the degree symbol "°" is not shown, our measure is given in terms of radians. Arc Length on a Circle
From our definition of our angle θ in radians above, we can obtain a formula for finding the length of an arc of a circle. We know that the measure of our angle θ in radians is represented by: $$θ=\frac{s}{r}$$ If we multiply both sides by r, we will obtain: $$\frac{s}{r}=θ$$ $$\require{cancel}\cancel{r}\cdot \frac{s}{\cancel{r}}=r \cdot θ$$ $$s=r θ$$ This leads us to the following formula: Formula for Arc Length
The length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by: $$s=rθ$$ where θ is in radians. Let's look at a few examples.Example #1: Find the length of each arc.
$$θ=\frac{π}{4}$$ $$r=12\hspace{.2em}\text{miles}$$ To find the length of our given arc, let's use our formula: $$s=r θ$$ Let's plug into our formula: $$s=12 \hspace{.2em}\text{miles}\cdot \frac{π}{4}$$ $$s=3\cancel{12}\hspace{.2em}\text{miles}\cdot \frac{π}{\cancel{4}}$$ $$s=3π \hspace{.2em}\text{miles}$$ Example #2: Find the length of each arc. $$θ=135°$$ $$r=12\hspace{.2em}\text{inches}$$ In this case, we can't immediately use our formula since our angle measure is given in degrees. We need to first convert θ to radians. $$135° \cdot \frac{π}{180°}=\frac{3π}{4}$$ Now that θ has been converted into radians, we can use our formula: $$s=r θ$$ $$s=12 \hspace{.2em}\text{inches}\cdot \frac{3π}{4}$$ $$s=3\cancel{12}\hspace{.2em}\text{inches}\cdot \frac{3π}{\cancel{4}}$$ $$s=9π \hspace{.2em}\text{inches}$$ How to Find the Area of a Sector of a Circle
We have already seen that a complete circle or one full rotation forms an angle with a measure of 360° or $2π$ radians. The formula for the area of a complete circle with a radius r: $$\text{area of a circle}=π r^2$$ A sector of a circle is the portion of the interior of a circle intercepted by a central angle. Visually, it looks like a "piece of pie". If a central angle for a sector of a circle has measure θ radians, then the sector makes up the fraction: $\frac{θ}{2π}$ of a complete circle. To obtain the area of a sector, we will modify our area of a circle formula: Area of a Sector of a Circle with Radius r and Central Angle θ Measured in Radians
$$\frac{θ}{2π}(πr^2)=\frac{1}{2}r^2 θ$$ Let's look at an example.Example #3: Find the area of each sector.
$$θ=\frac{3π}{2}$$ $$r=9 \hspace{.1em}\text{meters}$$ Let's plug into our formula: $$\frac{1}{2}r^2 θ$$ $$\frac{1}{2}\cdot (9 \hspace{.1em}\text{meters})^2 \cdot \frac{3π}{2}$$ $$\frac{243 π}{4}\hspace{.1em}\text{square meters}$$ Example #4: Find the area of each sector. $$θ=60°$$ $$r=13 \hspace{.1em}\text{yards}$$ Since θ is given in degrees, our first step is to convert θ into radians. $$60° \cdot \frac{π}{180°}=\frac{π}{3}$$ Now, we can plug into our formula: $$\frac{1}{2}r^2 θ$$ $$\frac{1}{2}\cdot (13 \hspace{.1em}\text{yards})^2 \cdot \frac{π}{3}$$ $$\frac{169 π}{6}\hspace{.1em}\text{square yards}$$ Skills Check:
Example #1
Find the length of each arc. $$r=15\text{m}$$ $$θ=\frac{4π}{3}$$
Please choose the best answer.
A
$$150 π \hspace{.2em}\text{m}$$
B
$$\frac{27}{8}π \hspace{.2em}\text{m}$$
C
$$60 π \hspace{.2em}\text{m}$$
D
$$\frac{40}{3}π \hspace{.2em}\text{m}$$
E
$$20 π \hspace{.2em}\text{m}$$
Example #2
Find the length of each arc. $$r=17\text{ft}$$ $$θ=\frac{3π}{4}$$
Please choose the best answer.
A
$$\frac{51π}{4}\hspace{.2em}\text{ft}$$
B
$$\frac{26π}{5}\hspace{.2em}\text{ft}$$
C
$$24π \hspace{.2em}\text{ft}$$
D
$$169π \hspace{.2em}\text{ft}$$
E
$$\frac{56}{3}\hspace{.2em}\text{ft}$$
Example #3
Find the area of each sector. $$r=9\text{m}$$ $$θ=255°$$
Please choose the best answer.
A
$$\frac{20π}{3}\hspace{.2em}\text{m}^2$$
B
$$\frac{39π}{8}\hspace{.2em}\text{m}^2$$
C
$$\frac{152π}{3}\hspace{.2em}\text{m}^2$$
D
$$\frac{459π}{8}\hspace{.2em}\text{m}^2$$
E
$$225π \hspace{.2em}\text{m}^2$$
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