1. What is a Segment of a Circle?
A segment of a circle is the region bounded by a chord and the arc between the chord’s endpoints.
- Minor segment: smaller region
- Major segment: larger region
Segments appear when a circle is cut by a chord, creating a curved region.
2. Understanding Segment Area
The area of a segment is found by subtracting the area of the triangle (formed by two radii and the chord) from the area of the sector.
A_{\text{segment}} = A_{\text{sector}} - A_{\triangle}
3. Formula for Area of a Minor Segment
If the central angle of the segment is \(\theta\) (in degrees), then:
A_{\text{segment}} = \dfrac{\theta}{360^\circ} \pi r^2 - A_{\triangle}
3.1. Triangle Area (Using Central Angle)
In a circle of radius \(r\) with central angle \(\theta\), the triangle formed is isosceles. Its area is:
A_{\triangle} = \dfrac{1}{2} r^2 \sin(\theta)
(Here, \(\theta\) is in radians for the sine formula.)
4. Area Formula Using Radians (Most Common in NCERT)
If \(\theta\) is in radians, the area of the segment is:
A_{\text{segment}} = \dfrac{1}{2} r^2 (\theta - \sin \theta)
This formula directly gives the area of a minor segment.
5. Major Segment Area
The major segment is the rest of the circle:
A_{\text{major segment}} = \pi r^2 - A_{\text{minor segment}}
6. Example Problems
Here are clear examples to understand segment area calculations:
6.1. Example 1: Using Degrees
A circle has radius \(r = 10\text{ cm}\) and a chord subtends \(\theta = 60^\circ\) at the centre.
Step 1: Sector area
A_{\text{sector}} = \dfrac{60}{360} \pi r^2 = \dfrac{1}{6} \pi \times 100 = \dfrac{100\pi}{6}
Step 2: Triangle area
A_{\triangle} = \dfrac{1}{2} r^2 \sin(60^\circ) = \dfrac{1}{2} \times 100 \times \dfrac{\sqrt{3}}{2} = 25\sqrt{3}
Step 3: Segment area
A_{\text{segment}} = \dfrac{100\pi}{6} - 25\sqrt{3}
6.2. Example 2: Using Radians
A circle has radius \(r = 7\text{ cm}\) and central angle \(\theta = \dfrac{\pi}{2}\) radians.
A_{\text{segment}} = \dfrac{1}{2} r^2 (\theta - \sin \theta) = \dfrac{1}{2} \times 49 \left( \dfrac{\pi}{2} - 1 \right)
= \dfrac{49}{2} \left( \dfrac{\pi}{2} - 1 \right) \text{ cm}^2
7. Where Segment Area is Used
Segment area is useful in:
- Architectural curved designs
- Road and bridge arch planning
- Water tank cross-section calculations
- Finding area cut off by chords in fields or circular plots