# Section 3.2

:::{prf:definition} Arc Length
:label: arcDef

The length $s$ of the arc intercepted on a circle of radius $r$
by a central angle of measure $\theta$ radians is given by the product
of the radius and the radians measure of the angle

$$
s=r\theta
$$

where $\theta$ **must** be in units of radians.
:::

:::{prf:definition} Sector Area
:label: sectorArea

The area $A$ of a sector of a circle of radius $r$ and central angle $\theta$ is given by

$$
A=\frac{1}{2}r^{2}\theta
$$

where $\theta$ is in radians.
:::

::::{prf:example}
:label: arcLengthExam

Find the arc length for the following situations (leave your answers in terms of $\pi$):

Circle has a radius of $5$ meters and $\theta=\dfrac{3\pi}{2}$.

:::{dropdown} Solution:

\begin{align*}
s & =r\theta\\
 & =5\cdot\frac{3\pi}{2}\\
 & =\dfrac{15\pi}{2}
\end{align*}

:::

Circle has a radius of $8.5$ meters and $\theta=210^{\circ}$.

:::{dropdown} Solution:

First, we will write $\theta$ in radians

$$
210\cdot\frac{\pi}{180}=\frac{7\,\pi}{6}=\theta
$$

Finally, we will calculate the arc length

\begin{align*}
s & =r\theta\\
 & =8.5\cdot\dfrac{7\pi}{6}\\
 & =\dfrac{17}{2}\cdot\dfrac{7\pi}{6}\\
 & =\frac{119\pi}{12}
\end{align*}

:::

Let Gear 1 connect to Gear 2 where if Gear 1 turns, then Gear 2 turns with it. If Gear 1 has a radius of 2 meters and rotates $45^{\circ}$ and Gear 2 has a radius of $3$ meters, then find how many degrees does Gear 2 rotate.

:::{dropdown} Solution:

First, we will find $45^{\circ}$ in radians

$$
45\cdot\frac{\pi}{180}=\frac{\pi}{4}=\theta_{\text{Gear 1}}
$$

The arc length for gear 1 is

\begin{align*}
s_{\text{Gear 1}} & =r_{\text{Gear 1}}\cdot\theta_{\text{Gear 1}}\\
 & =2\cdot\dfrac{\pi}{4}\\
 & =\dfrac{\pi}{2}
\end{align*}

Since gear 1 and gear 2 travel the same arc length we know $s_{\text{Gear 2}}=\frac{\pi}{2}$. Furthermore, since $r_{\text{Gear 2}}=3$ we know

\begin{align*}
s_{\text{Gear 2}} & =r_{\text{Gear 2}}\cdot\theta_{\text{Gear 2}}\\
\frac{\pi}{2} & =3\theta_{\text{Gear 2}}\\
\theta_{\text{Gear 2}} & =\dfrac{\pi}{6}=30^{\circ}
\end{align*}

Thus, gear 2 rotates $30^{\circ}$ as gear 1 rotates $45^{\circ}$.
:::

::::

::::{prf:example}
:label: secAreaExam

Find the sector area for the following situation (leave your answers in terms of $\pi$):

A pizza is sliced into 9 pieces equal slices. If you order a 16 inch pizza, then what is the area of one slice of the pizza?

:::{dropdown} Solution:

A circle is made up of $360^{\circ}$ and if the circle is partitioned
into 9 equal pieces than $360^{\circ}$ is divided into 9 equal pieces:

$$
\frac{360}{9}=40.
$$

Next, we want to find $40^{\circ}$ in units of radians

$$
40\cdot\frac{\pi}{180}=\frac{2\,\pi}{9}
$$

The measure of ``16 inch pizza'' is stating the diameter of the circle is 16 inches; therefore, the radius of the this pizza is $\frac{16}{2}=8$.

The area of a sector is $A=\frac{1}{2}r^{2}\theta$, where in this case $\theta=\frac{2\pi}{9}$ and $r=8$. Therefore,

\begin{align*}
A & =\frac{1}{2}(8)^{2}\cdot\frac{2\pi}{9}\\
 & =\frac{64\pi}{9}
\end{align*}

:::
::::