Sector and Segment

Tags

math

• An arc is a section of the circumference of the circle
• A sector is an area enclosed by two radii and an arc
• A chord is a straight line connecting two points on the circumference of a circle
• A segment is a section formed between an arc and a chord

\begin{figure}[ht]
  \centering
  \incfig{20210609175724}
  \caption{Annotation of a circle with both a segment and a sector.}
\end{figure}

Sector Area

\begin{align} \label{eq:sector-area-radians} a = \frac{r^2 \theta^{c}}{2} \end{align}

The area of a sector in degrees is the ratio of the area taken up by the sector compared to the rest of the circle. This is a ratio in terms of the total area of the circle.

\begin{align} \label{eq:sector-area-proof} A &= \frac{\theta^{\degree}}{2 \pi} \times \pi \times r^2 \\

&= \frac{\theta^c}{2} \times r^2

\end{align}

In eq:sector-area-proof we substitute the equation for the conversion between degrees and radians and find that the \( \pi \) cancels out leaving us with eq:sector-area-radians.

Arc Length

\begin{align} \label{eq:arc-length-radians} l_\text{arc} = r \theta^c \end{align}

Refers to the circumference of a sector. This is calculated in the same way as sector area, as a ratio of the circumference of the full circle and the angle of the sector taken up by the segment.

\begin{align} \label{eq:arc-length-proof} l_\text{arc} &= \theta^{\degree} \frac{2 \pi}{360} r \\

           &= \theta^c r

\end{align}

Recall that the circumference of a circle is \( 2 \pi r \). If we substitute \( 2\pi \) for an arc around the entire circle into eq:arc-length-proof we get back the original circumference equation.

Segment Area

\begin{figure}[ht]
  \centering
  \incfig{20210609195028}
\end{figure}

Observe that a segment is just the region unoccupied by the triangle connecting the center of the circle to the segment. We can calculate the area of this triangle after finding its height and then subtract this from the total area of the sector to get the segment area.

\begin{align} \frac{r^2 [\theta^c - \sin \theta]}{2} \end{align}