Areas Related to Circles | Class 10 Mathematics Notes

Deriving the Formula for Area of a Sector and Length of an Arc

This section focuses on finding formulas to calculate the area of a sector and the length of the arc corresponding to that sector. Consider a circle with center O and radius r. Let the sector be formed by two radii OA and OB enclosing an angle θ degrees at the center (see Fig. 11.3).

We know the area of the entire circle is πr², which corresponds to a central angle of 360°. Using the unitary method, the area corresponding to 1° is (πr²)/360. Therefore, the area of the sector with angle θ is (θ/360) × πr². This formula allows us to calculate the area of any sector by knowing the radius and the angle it subtends at the center.

Similarly, the length of the entire circumference is 2πr, corresponding to 360°. The length of the arc corresponding to angle θ is (θ/360) × 2πr. This formula helps us find the length of any arc when the radius and central angle are known.

These formulas are fundamental in geometry and have practical applications in fields such as engineering, architecture, and design, where parts of circles are common.

📊 Diagram: Fig. 11.3 illustrates sector OAPB with radius r and central angle θ; Fig. 11.4 shows arc APB corresponding to sector with angle θ.

🧪 Activity: Derive area and arc length formulas using the unitary method.

🔗 Connection: Leads to applying these formulas in calculating areas of segments and solving related problems.