Areas Related to Circles | Class 10 Mathematics Notes

Area of a Segment of a Circle

A segment of a circle is the region bounded by a chord and the corresponding arc. To find the area of a segment, we use the relationship between the area of the sector formed by the two radii and the chord, and the area of the triangle formed by the two radii and the chord.

Consider the segment APB formed by chord AB in a circle with center O and radius r (see Fig. 11.4). The area of the segment APB is equal to the area of the sector OAPB minus the area of triangle OAB.

Mathematically, Area of segment = (θ/360) × πr² – Area of ΔOAB, where θ is the central angle subtended by the chord AB.

This formula is practical because while the area of the sector is straightforward to calculate using the earlier formula, the area of the triangle can be found using trigonometric methods or coordinate geometry depending on the given data.

Additionally, the major segment's area can be found by subtracting the minor segment's area from the total circle area πr². This approach helps in solving problems involving segments, such as calculating the area of a shaded region bounded by a chord and arc.

📊 Diagram: Fig. 11.4 shows segment APB formed by chord AB and arc APB in circle with center O and radius r.

🧪 Activity: Calculate segment areas by subtracting triangle areas from sector areas.

🔗 Connection: Prepares for solving numerical problems involving sectors and segments in the next section.