Circular Sector

A circular sector is a slice of a circle enclosed by two radii and the arc connecting them.
the circular sector

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.

     

     
     

    Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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