Geometry - Sector Of Circle

Sector of Circle

Understanding a Sector of a Circle: Definition, Properties, and Key Formulas

A sector of a circle is a portion of a circle enclosed by two radii and the arc between them. It resembles a "slice of pie" or a "wedge" and is commonly seen in pie charts, clocks, and engineering components involving rotation. The size of the sector depends on the central angle subtended at the center.

Diagram showing a sector of a circle with radius, arc, and central angle.

Key Properties of a Sector

  1. Defined by Two Radii and an Arc: It includes the central angle and a curved boundary.
  2. Measured in Degrees or Radians: The central angle can be expressed in either, depending on the formula used.
  3. Part of a Circle: A full circle is a sector with a \(360^\circ\) angle.

Key Formulas for a Sector

1. Area \(A\) of a Sector:

The area of a sector depends on the central angle and radius:

\[ A = \frac{\pi r^2 \alpha}{360} = \frac{br}{2} \]

where:

  • \(r\): Radius of the circle
  • \(\alpha\): Central angle in degrees
  • \(b\): Length of the arc (see below)

2. Arc Length \(b\):

This formula calculates the curved boundary of the sector:

\[ b = \frac{2 \pi r \alpha}{360} \]

where:

  • \(r\): Radius of the circle
  • \(\alpha\): Central angle in degrees

Applications of Circle Sectors

  • Statistics and Data Visualization: Used in pie charts and circular diagrams.
  • Engineering: Applied in gear design, turbine blades, and rotational mechanics.
  • Architecture and Design: Seen in arcs, fan-shaped elements, and decorative patterns.
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