Class 10 NCERT Maths Chapter 11: Areas Related to Circles
In this chapter, we learn to calculate areas of circular regions and their parts such as sectors, segments, and combinations with other shapes. These concepts are useful in solving practical problems involving design, construction, and measurement.
1. Perimeter and Area of a Circle
Formulas:
- Circumference: \(C = 2 \pi r\)
- Area: \(A = \pi r^2\)
Here, \(r\) is the radius of the circle and \(\pi\) is taken as \(\frac{22}{7}\) or \(3.14\) unless specified otherwise.
2. Area of a Sector
A sector is the portion of a circle enclosed by two radii and the corresponding arc.
| Quantity | Formula |
|---|---|
| Area of sector (angle \(\theta\) in degrees) | \(\frac{\theta}{360} \times \pi r^2\) |
| Arc length | \(\frac{\theta}{360} \times 2\pi r\) |
3. Area of a Segment
A segment is the region between a chord and the corresponding arc of a circle.
Formula:
\(\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}\)
The area of the triangle depends on the given information (it may be an isosceles triangle when the radii are equal).
4. Problems on Combined Figures
Often, circles or their parts are combined with other shapes like rectangles, triangles, or squares. In such cases:
- Find the required area of each part separately.
- Add or subtract areas as per the figure.
Example: Finding the area of a flower bed in the shape of a ring (difference of areas of two circles).
\[ \text{Area of ring} = \pi R^2 - \pi r^2 = \pi (R^2 - r^2) \] where \(R\) = outer radius and \(r\) = inner radius.
Key Points to Remember
- Always keep units consistent (e.g., all lengths in cm, area in cm\(^2\)).
- Use \(\pi = \frac{22}{7}\) for exact fractional values, and \(\pi = 3.14\) for decimal approximation unless instructed otherwise.
- Draw neat diagrams to understand the figure before applying formulas.