Circular Sector
A circular sector is a slice of a circle enclosed by two radii and the arc connecting them.
The central angle, defined by these two radii, dictates the sector's size.
An arc is a segment of the circle's perimeter, bordered by the radii.
The central angle is the angle between the sector's radii, whereas a radius is a line stretching from the center of the circle to any point on the circumference.
A circular segment with a single base refers to the region between an arc and its subtending chord.
Key Properties and Formulas
The length (L) of an arc within a sector can be determined with the formula:
$$ L = \frac{\alpha}{360} \cdot 2\pi r = \frac{\alpha}{180} \cdot \pi r $$
Where α is the central angle in degrees, and r denotes the circle's radius.
The area (As) of a sector is calculated using:
$$ A_s = \frac{\alpha}{360} \cdot \pi r^2 $$
For α in radians, the sector's area formula transforms to:
$$ A_s = \frac{\alpha_{rad}}{2} \cdot r^2 $$
Proof. The circular sector's area is proportional to the central angle $$ A_s : A = \alpha:360 $$ This means the sector's area As is to the circle's total area A as the central angle α is to a complete rotation (360°). Represented as a fraction, $$ \frac{A_s}{A} = \frac{\alpha}{360} $$ Dividing both sides by A yields $$ A_s = \frac{\alpha}{360} \cdot A $$ Given the circle's area is A=πr2, we derive $$ A_s = \frac{\alpha}{360} \cdot \pi r^2 $$ To express this in radians, recognizing that a full rotation (360°) equals 2π radians simplifies to $$ A_s = \frac{\alpha_{rad}}{2 \pi} \cdot \pi r^2 = \frac{\alpha_{rad}}{2} \cdot r^2 $$
Alternatively, the sector's area can be derived from the arc length when the angle is given in degrees.
$$ A_s = \frac{1}{2} \cdot L \cdot r $$
Proof. Starting from the degree-based area formula for a sector, $$ A_s = \frac{\alpha}{360} \cdot \pi r^2 $$ Dividing both terms by L results in $$ \frac{ A_s }{L} = \frac{\alpha}{360} \cdot \pi r^2 / L $$ Given the arc length formula $ L = \frac{\alpha}{360} \cdot 2\pi r $, simplification leads to $$ \frac{ A_s }{L} = \frac{r }{2 } $$ Thus, we arrive at $$ A_s = \frac{1}{2} \cdot L \cdot r $$
And so forth.