Class 10 Mathematics
Sector & Segment
⏱ 12 min read
In this chapter, we will explore the concepts of sectors and segments in circles. These two geometric shapes are essential in understanding various applications in mathematics, particularly in geometry and trigonometry. By the end of this chapter, you will be able to identify, define, and calculate the area and perimeter of sectors and segments of circles.
A circle is a closed shape where all points are equidistant from a fixed point called the center. The distance from the center to any point on the circle is known as the radius (r). The total distance around the circle is called the circumference (C), calculated using the formula:
C = 2πr
where π (pi) is approximately 3.14.
A sector is a portion of a circle enclosed by two radii and the arc between them. It resembles a 'slice' of pizza.
The area (A) of a sector can be calculated using the formula:
A = (θ/360) × πr²
where θ is the angle in degrees of the sector.
Example: If the radius of a circle is 10 cm and the angle of the sector is 60°, then the area of the sector is:
A = (60/360) × π × (10)² = (1/6) × π × 100 = (100π/6) cm² ≈ 52.36 cm².
The perimeter (P) of a sector includes the lengths of the two radii and the arc length (L). It can be calculated as:
P = 2r + L
The arc length (L) can be calculated using:
L = (θ/360) × 2πr
A segment is the area enclosed by a chord and the arc that subtends it. It can be visualized as the 'cap' of a pizza slice.
To find the area of a segment, you first need to calculate the area of the sector and then subtract the area of the triangle formed by the two radii and the chord.
Area of Triangle = (1/2) × base × height
For a triangle formed in a segment, the base is the chord and the height can be found using trigonometric ratios or specific formulas depending on the angle.
Example: If the radius is 10 cm and the angle is 60°, the area of the sector is 52.36 cm² as calculated earlier. If the area of the triangle formed is 25 cm², then the area of the segment is:
Area of Segment = Area of Sector - Area of Triangle = 52.36 cm² - 25 cm² = 27.36 cm².
Understanding sectors and segments is crucial in various fields, including architecture, engineering, and design. They help in calculating areas needed for construction, designing circular objects, and more.
In this chapter, we learned about sectors and segments of circles. We defined both terms and explored their properties. We also learned how to calculate the area and perimeter of sectors and the area of segments. These concepts are fundamental in geometry and have practical applications in real-world scenarios. Understanding these shapes will aid in solving more complex mathematical problems involving circles.
🧠 Ready to test your knowledge?
Take the quiz for this chapter to reinforce what you just learned and track your progress.