How Do You Find the Area of a Sector?

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Why the Formula Works

A sector is the fraction of a circle enclosed by two radii and their connecting arc. A complete circle has area $\pi r^2$ and central angle $2\pi$ radians, so a sector with angle $\theta$ occupies the fraction

of the complete circle. Therefore,

This derivation explains why the radius is squared and why the radian formula contains no additional factor of $\pi$. Because sector area is directly proportional to the central angle, doubling $\theta$ while $r$ remains fixed doubles the area.

Worked Example

A sector has radius $6\text{ cm}$ and central angle $\frac{2\pi}{3}$ radians. Since the angle is already in radians, substitute directly:

The exact area is $12\pi\text{ cm}^2$, or approximately $37.7\text{ cm}^2$ to three significant figures.

A useful check is to compare the answer with the full circle area. Here, the circle has area $36\pi\text{ cm}^2$, and the angle $\frac{2\pi}{3}$ is one third of $2\pi$. The sector should therefore have area $\frac{1}{3}(36\pi)=12\pi\text{ cm}^2$, confirming the calculation. This proportional check is especially valuable on Paper 1 because it can expose an incorrect angle unit.

If the angle were instead $120^\circ$, use the degree formula. Alternatively, convert $120^\circ$ to $\frac{2\pi}{3}$ radians; both methods produce the same result.

The common misconception is using $r\theta$ for area. That formula calculates arc length, whereas sector area requires $\frac{1}{2}r^2\theta$. Never insert a degree value directly into the radian formula.

IB Exam Technique

This is IB Mathematics: Analysis and Approaches, SL 3.4, core content for SL and HL. It may appear on Paper 1 or Paper 2. Radians are assumed unless degrees are stated. Show the formula and substitution, retain exact answers involving $\pi$ when appropriate, round decimals to three significant figures unless instructed otherwise, and include squared units such as $\text{cm}^2$.