Sector Area
The 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, bounded by two radii and the arc between them, and its area is a fraction of the whole circle: $A = (\theta/360)\pi r^2$ for central angle $\theta$ in degrees, or $A = (1/2)r^2\theta$ in radians. When $\theta = 360$ degrees ($2\pi$ radians), $A = \pi r^2$, the full circle. This is derivable by integration in polar coordinates, $$A = \frac{1}{2}\int_0^\theta r^2 \, d\varphi = \frac{1}{2}r^2\theta,$$ and for a general polar curve $r = f(\varphi)$ the enclosed area becomes $A = (1/2)\int f(\varphi)^2 \, d\varphi$.
Example
A circle with radius $6$ cm and a central angle of $90$ degrees gives a quarter-circle sector: $A = (90/360) \times \pi \times 6^2 = (1/4)(113.1) = 28.3$ cm$^2$. A sector with $r = 10$ m and $\theta = 150$ degrees gives $A = (150/360)\pi \cdot 100 = 130.9$ m$^2$, matching the radian calculation $\theta = 5\pi/6$, $A = (1/2)(100)(5\pi/6) = 130.9$ m$^2$. For the cardioid $r = a(1 + \cos\varphi)$, 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
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 $1/6$, so you just multiply the fraction by the full circle area. The sector area formula $A = (1/2)r^2\theta$ in radians directly parallels the triangle area formula $A = (1/2)bh$, treating $r$ as both base and height of a thin triangular slice, which reveals why radians are the natural angle unit; the polar area formula $A = (1/2)\int r^2 \, d\varphi$ generalizes this to unify the measurement of spiral, circular, and petal-shaped regions under one integral.