Areas Related to Circles

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.

• Sector is the region enclosed by two radii and the arc between them.
• Segment is the region enclosed by a chord and the corresponding arc.
• Each pair of radii or chord divides the circle into minor and major sectors or segments.
• The angle of the sector is the angle formed at the center by the two radii.
• Minor sector/segment refers to the smaller part, major sector/segment to the larger part.
• Understanding sectors and segments is foundational for calculating areas related to circles.
• 📌 Sector: Part of a circle enclosed by two radii and the arc between them.
• 📌 Segment: Part of a circle enclosed by a chord and the corresponding arc.
• 📌 Minor sector/segment: Smaller part formed by radii or chord.

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.

• Area of full circle = πr² corresponds to 360° angle.
• Area of sector with angle θ = (θ/360) × πr².
• Circumference of circle = 2πr corresponds to 360° angle.
• Length of arc with angle θ = (θ/360) × 2πr.
• Unitary method is used to derive these formulas.
• These formulas are essential for calculating areas and lengths related to parts of circles.
• 📌 Radius (r): Distance from center to any point on the circle.
• 📌 Central angle (θ): Angle subtended at the center by two radii.
• 📌 Arc length: Distance along the curved line of the sector.