Chapter 11: Areas Related to Circles
Understanding Sectors and Segments
A circle can be divided into parts called sectors and segments. A sector is the region enclosed by two radii and the arc between them, while a segment is the region between a chord and its corresponding arc. When the central angle is smaller, it forms a minor sector or segment; when it is larger, it forms a major sector or segment. The area of a sector with radius r and angle θ (in degrees) is given by:
Area of sector of angle θ =θ 360×πr²
Length of arc of a sector of angle θ = θ 360× 2πr
Calculating Segment Areas
The area of a segment is found by subtracting the area of the triangle formed by the radii and the chord from the area of the corresponding sector:
Area of segment=Area of sector−Area of triangle
To find the triangle’s area, trigonometric ratios such as sine and cosine may be used, especially when the angle at the center is known. The segment area depends on the height from the center to the chord and the length of the chord, which can be derived using symmetry and right triangle concepts.
Applications and Problem Solving
This chapter presents real-life applications involving partial circles—like the area swept by clock hands, the grazing area for a tied animal, or decorative circular designs. Problems involve calculating the area of circular sectors, segments, and the increase in area when the radius changes. Students apply formulas using given angles and radii, often involving values of π as 3.14 or 22/7, and approximate values for square roots as needed in trigonometric calculations.