Area of a Circular Sector
What is a Circular Sector?
A sector of a circle is the region enclosed between two radii and the arc connecting them. Think of it like a "pizza slice" — the tip is at the center of the circle, and the curved edge is part of the circumference.
Deriving the Formula
The key idea is that a sector is a fraction of the entire circle. The fraction is determined by how much of the full rotation ($360°$ or $2\pi$ radians) the central angle $\theta$ covers.
Step 1: Area of a Full Circle
$$A_{\text{circle}} = \pi r^2$$
Step 2: What Fraction of the Circle is the Sector?
If the central angle is $\theta$ (in degrees), the fraction of the circle is:
$$\text{Fraction} = \frac{\theta}{360°}$$
Step 3: Multiply to Get the Sector Area
Formula (Degrees):
$$A_{\text{sector}} = \frac{\theta}{360°} \times \pi r^2$$
Formula (Radians):
$$A_{\text{sector}} = \frac{1}{2} r^2 \theta$$
where $\theta$ is in radians
Why does the radian formula work?
Since a full circle is $2\pi$ radians, the fraction becomes $\frac{\theta}{2\pi}$. So:
$$A = \frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2}r^2\theta$$