Section 3.2

Definition 8 (Arc Length)

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.

Definition 9 (Sector Area)

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.

Example 19

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

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

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 \(\theta=210^{\circ}\).

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.

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 \(45^{\circ}\).

Example 20

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?

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,

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