Areas Related to Circles | Class 10 Mathematics Notes

11.1 Areas of Sector and Segment of a Circle

In this section, we revisit and deepen our understanding of two important parts of a circle: the sector and the segment. A sector of a circle is the region enclosed by two radii and the arc between them, essentially a 'slice' of the circle. The angle formed at the center of the circle by these two radii is called the angle of the sector. For example, if the circle's center is O and the radii are OA and OB, then the sector is the region OAPB, and the angle ∠AOB is the sector's angle. There are two sectors formed by any two radii: the minor sector (smaller area) and the major sector (larger area). The major sector's angle is 360° minus the minor sector's angle.

A segment of a circle is the region bounded by a chord and the corresponding arc. If AB is a chord, then the segment is the region APB enclosed between the chord AB and the arc APB. Similar to sectors, there are minor and major segments formed by the chord.

Understanding these parts of a circle is crucial because they appear frequently in geometry problems and real-world applications such as designing circular fields, wheels, and arcs in architecture. The section sets the stage for deriving formulas to calculate the areas of these sectors and segments, which are essential for solving practical problems involving parts of circles.

📊 Diagram: Fig. 11.1 shows a circle with center O, two radii OA and OB enclosing the minor sector OAPB and major sector OAQB; Fig. 11.2 shows a circle with chord AB forming minor segment APB and major segment AQB.

🧪 Activity: Recall and identify sectors and segments in given circle diagrams.

🔗 Connection: This section leads to the derivation of formulas for the areas of sectors and segments, and the length of arcs, which are covered in the next section.