# Area of Sector

## Theorem

Let $\mathcal C = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$.

Let $BAC$ be the sector of $\mathcal C$ whose angle between $AB$ and $AC$ is $\theta$.

Then the area $\mathcal A$ of sector $BAC$ is given by:

- $\mathcal A = \dfrac {r^2 \theta} 2$

where:

## Proof 1

Let $\mathcal C$ be a circle of radius $r$ whose center $A$ is at the origin and with radii $AB$ and $AC$.

Let $B$ and $C$ be arbitrary points on the circumference of $\mathcal C$.

Let $C$ be positioned at $\tuple {0, r}$ so that $AC$ coincides with the $y$-axis.

Let $BAC$ be the sector of $\mathcal C$ whose angle between $AB$ and $AC$ is $\theta$.

We have that $\mathcal C$ is centered at the origin.

From Equation of Circle: Corollary 2, the equation of $\mathcal C$ is given by:

- $r^2 = x^2 + y^2$

Let $B$ have co-ordinates:

- $\paren {x_0, y_0} = \paren {x_0, \sqrt {r^2 - {x_0}^2} }$

Triangle $\triangle ADB$ is a right triangle whose leg $DB$ is opposite $\theta$, and whose hypotenuse is $AB$.

Hence:

\(\displaystyle \sin \theta\) | \(=\) | \(\displaystyle \frac {\size {BD} } {\size {AB} }\) | $\quad$ Definition of Sine | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {x_0} r\) | $\quad$ Definition of $x_0$ and $r$ | $\quad$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \map \arcsin {\sin \theta}\) | \(=\) | \(\displaystyle \map \arcsin {\frac {x_0} r}\) | $\quad$ taking $\arcsin$ of both sides | $\quad$ | ||||||||

\((1):\quad\) | \(\displaystyle \leadsto \ \ \) | \(\displaystyle \theta\) | \(=\) | \(\displaystyle \map \arcsin {\frac {x_0} r}\) | $\quad$ Definition of Arcsine | $\quad$ |

Consider the line $AB$, which by definition passes through $A = \paren {0, 0}$ and $B = \paren {x_0, \sqrt {r^2 - {x_0}^2} }$.

Let:

- $\Delta y$ be the change in $y$ between $A$ and $B$
- $\Delta x$ be the change in $x$
- $m$ be the slope of $AB$.

\(\displaystyle \Delta y\) | \(=\) | \(\displaystyle \sqrt {r^2 - {x_0}^2} - 0\) | $\quad$ Difference between $y$ values of $A$ and $B$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sqrt {r^2 - {x_0}^2}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \Delta x\) | \(=\) | \(\displaystyle x_0 - 0\) | $\quad$ Difference between $x$ values of $A$ and $B$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle x_0\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle m\) | \(=\) | \(\displaystyle \frac {\Delta y} {\Delta x}\) | $\quad$ Definition of Slope of Straight Line | $\quad$ | ||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {\sqrt {r^2 - {x_0}^2} } {x_0}\) | $\quad$ | $\quad$ | |||||||||

\((2):\quad\) | \(\displaystyle y\) | \(=\) | \(\displaystyle \frac {\sqrt {r^2 - {x_0}^2} } {x_0} x\) | $\quad$ Equation of Straight Line in Plane in Gradient-Intercept Form: $y$ intercept is $0$ | $\quad$ |

Let $K$ be the region between $\mathcal C$, $AC$ and $AB$, from $0$ to $x_0$, coloured orange in the diagram above.

This is the sector of $\mathcal C$ whose area we are to find.

Let $\mathcal A$ be the area of $K$.

Then:

\(\displaystyle \mathcal A\) | \(=\) | \(\displaystyle \int_0^{x_0} \paren {\sqrt {r^2 - x^2} - \frac {\sqrt {r^2 - {x_0}^2} } {x_0} x} \rd x\) | $\quad$ integrating between two curves to find area | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_0^{x_0} \sqrt {r^2 - x^2} \rd x - \frac {\sqrt {r^2 - {x_0}^2} } {x_0} \int_0^{x_0} x \rd x\) | $\quad$ Linear Combination of Integrals | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_0^{x_0} \sqrt {r^2 - x^2} \rd x - \frac {\sqrt {r^2 - {x_0}^2} } {x_0} \sqbrk {\frac {x^2} 2}_0^{x_0}\) | $\quad$ Primitive of Power | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sqbrk {\frac {x \sqrt {r^2 - x^2} } 2 + \frac {r^2} 2 \arcsin \frac x r}_0^{x_0} - \frac {\sqrt {r^2 - {x_0}^2} } {x_0} \sqbrk {\frac {x^2} 2}_0^{x_0}\) | $\quad$ Primitive of $\sqrt {a^2 - x^2}$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {x_0 \sqrt {r^2 - {x_0}^2} } 2 + \frac {r^2} 2 \arcsin \frac {x_0} r - \frac {\sqrt {r^2 - {x_0}^2} } {x_0} \frac { {x_0}^2} 2\) | $\quad$ evaluating over range $\closedint 0 {x_0}$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {x_0 \sqrt {r^2 - {x_0}^2} } 2 + \frac {r^2} 2 \arcsin \frac {x_0} r - \frac {x_0 \sqrt {r^2 - {x_0}^2} } 2\) | $\quad$ simplifying | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {r^2} 2 \arcsin \frac {x_0} r\) | $\quad$ cancelling | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {r^2 \theta} 2\) | $\quad$ from $(1)$ | $\quad$ |

$\blacksquare$

## Proof 2

From Area of Circle, the area of $\mathcal C$ is $\pi r^2$.

From Full Angle measures $2 \pi$ Radians, the angle within $\mathcal C$ is $2 \pi$.

The fraction of the area of $\mathcal C$ within the sector $BAC$ is therefore $\pi r^2 \times \dfrac \theta {2 \pi}$.

Hence the result.

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 4$: Geometric Formulas: $4.13$

- Weisstein, Eric W. "Circular Sector." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/CircularSector.html