Geometry - Segment Of Circle

Segment of Circle

Understanding the Segment of a Circle: Definition, Properties, and Area Formula

A segment of a circle is a region bounded by an arc and the chord connecting the arc's endpoints. Unlike a sector (which includes the center), a segment only captures the curved "slice" of the circle without extending to the center. Segments are commonly seen in circular windows, arches, and mechanical components.

Diagram of a segment of a circle showing radius and angle.

Key Properties of a Segment

  1. Defined by a Chord and Arc: A segment is formed by drawing a chord across a circle and considering the area between the chord and the arc.
  2. Two Types: Minor segment (smaller area) and major segment (larger area).
  3. Dependent on Angle: The size of the segment is based on the central angle subtended by the arc.

Key Formula for Area

Area \(A\) of a Segment:

This formula calculates the area enclosed by an arc and a chord:

\[ A = \frac{r^2}{2} \left( \frac{\pi \alpha^\circ}{180^\circ} - \sin(\alpha^\circ) \right) \]

where:

  • \(r\): Radius of the circle
  • \(\alpha^\circ\): Central angle in degrees
  • \(\pi \approx 3.14\)

Applications of Circle Segments

  • Architecture: Used in domes, arches, and bridges for aesthetic curves and structural calculations.
  • Engineering: Crucial in gear design and stress distribution analysis.
  • Geometry Problems: Common in advanced math problems involving partial areas.
×

×