How do you find the area of a sector?

In geometry, a sector is the portion of a circle enclosed by two radii and the arc that connects them. Think of the two radii as the straight cuts made from the center of a pizza to the crust, and the arc as the crust itself. The angle between those two straight cuts at the center is called the central angle, often represented by the Greek letter theta ($\theta$).

If your central angle $\theta$ is given in degrees, you compare it to the full 360 degrees of a circle. The fraction of the circle you have is $\frac{\theta}{360}$. To find the sector's area, you multiply this fraction by the total area of the circle. The precise formula is: $\text{Area} = \frac{\theta}{360} \times \pi r^2$.

In advanced math, angles are often measured in radians instead of degrees. A full circle is $2\pi$ radians. The fraction of the circle is therefore $\frac{\theta}{2\pi}$. When you multiply this by the total area $\pi r^2$, the $\pi$ cancels out, leaving a very clean formula: $\text{Area} = \frac{1}{2} r^2 \theta$. Remember, this only works if $\theta$ is in radians!

The most common mistake is using the wrong formula for the units given. If you plug an angle in degrees into the radian formula, your answer will be completely wrong. Always check whether your angle has a little degree symbol ($^\circ$). Another frequent error is forgetting to square the radius before multiplying, or accidentally squaring the entire $\pi r$ term instead of just $r$.