Sector Area
Geometry & MeasurementThe area of a sector is the region bounded by two radii and an arc, equal to a fractional part of the full circle's area.
Formula
A = \frac{\theta}{360}\pi r^2 \text{ or } A = \frac{1}{2}r^2\theta \text{ (radians)}
Definition
A sector of a circle looks like a slice of pie: two straight sides going to the center and a curved edge along the rim. The sector area is the area of that "pie slice."
Example
A circle with radius $6$ cm and a central angle of $90$ degrees: the sector is a quarter of the whole circle. Area $= (90/360) \times \pi \times 6^2 = (1/4)(113.1) = 28.3$ cm$^2$.
Key Insight
Think of a sector as a fraction of the whole circle. A $90$-degree sector is $1/4$ of the circle, a $180$-degree sector is $1/2$, and a $60$-degree sector is $1/6$. Just multiply the fraction by the full circle area.
Definition
The area of a sector with radius $r$ and central angle $\theta$ (in degrees) is $A = (\theta/360)\pi r^2$. In radians, $A = (1/2)r^2\theta$. A sector is bounded by two radii and the arc they intercept. When $\theta = 360$ degrees ($2\pi$ radians), $A = \pi r^2$, the full circle.
Example
A sector with $r = 10$ m and $\theta = 150$ degrees: $A = (150/360)\pi \cdot 100 = (5/12)\pi \cdot 100 = 130.9$ m$^2$. The same sector in radians: $\theta = 5\pi/6$, $A = (1/2)(100)(5\pi/6) = 130.9$ m$^2$.
Key Insight
The sector area $A = (1/2)r^2\theta$ in radians directly parallels the triangle area formula $A = (1/2)bh$ if you think of $r$ as both the base and height of a thin triangular slice. This reveals why radians are the natural angle unit.
Definition
The sector area is $A = (1/2)r^2\theta$, derivable by integration in polar coordinates: $$A = \frac{1}{2}\int_0^\theta r^2 \, d\varphi = \frac{1}{2}r^2\theta$$ for a circle of constant radius $r$. For a general polar curve $r = f(\varphi)$, the enclosed area is $A = (1/2)\int f(\varphi)^2 \, d\varphi$.
Example
For the cardioid $r = a(1 + \cos\varphi)$ in polar coordinates, the total area is $$A = \frac{1}{2}\int_0^{2\pi} a^2(1+\cos\varphi)^2 \, d\varphi = \frac{3}{2}\pi a^2.$$
Key Insight
The polar area formula $A = (1/2)\int r^2 \, d\varphi$ is the natural generalization of sector area to any polar curve, unifying the measurement of spiral, circular, and petal-shaped regions under one elegant integral.