Sector Area

Intuition and core definition

A sector is the region bounded by two radii and their intercepted arc. Its area is the same fraction of full circle area as its central angle is of a full turn: $A_s=(\theta/360^\circ)\pi r^2$, or $A_s=\frac12r^2\theta$ when $\theta$ is in radians.

Notation, language, and conditions

$r>0$ and $\theta$ must be a central angle. Degree and radian formulas are not interchangeable without conversion. A minor sector corresponds to an angle below $180^\circ$; a major sector uses the remaining angle $360^\circ-\theta$.

Why this idea matters

Sector area scales the full circle's area by the central angle's fraction of one revolution.

A dependable method

1. Identify radius, central angle, and angle units.
2. Choose the degree-fraction or radian formula.
3. Calculate the turn fraction and multiply by $\pi r^2$.
4. Keep an exact $\pi$ form before rounding.
5. Use square units and compare with the entire disk area.

Worked example

Common mistakes