Sectors & Segment Area

Sector area: take a fraction of the circle

The formula is $A = \dfrac{\theta}{360}\pi r^2$, where $\theta$ is the central angle in degrees and $r$ is the radius. Reduce the fraction first — a $90^\circ$ sector is $\displaystyle \frac{1}{4}$ of the circle, a $60^\circ$ sector is $\displaystyle \frac{1}{6}$, a $120^\circ$ sector is $\displaystyle \frac{1}{3}$. Then multiply by $\pi r^2$.

The same fraction gives arc length if you multiply by the circumference instead: arc length $\displaystyle = \frac{\theta}{360} \cdot 2\pi r$. Fraction times circumference for length, fraction times area for area — the setup is identical.

Segment area: sector minus triangle

Draw the chord connecting the two endpoints of a sector's arc. The chord splits the sector into a triangle (formed by the two radii and the chord) and a segment (the curved sliver between the chord and the arc).

So the plan is always: segment area $=$ sector area $-$ triangle area. The triangle's two known sides are both radii. When the central angle is $90^\circ$, the triangle is right with legs $r$ and $r$, so its area is $\displaystyle \frac{1}{2}r^2$ — the most common case in practice.

In the figure, the two radii and the chord $\overline{AB}$ form the triangle; the sliver between the chord and the arc is the segment.

Exact answers vs. decimals

Leave $\pi$ in the answer unless the problem says to round. An exact answer like $6\pi$ is cleaner and can't pick up rounding error. If a decimal is requested, substitute $\pi \approx 3.14159$ at the very last step, not earlier.

Worked examples

Try one yourself

Common questions