Areas Related to Circles Class 10 ||Maths|| Chapter 11 NCERT Notes

Areas Related To Circles Class 10 ||Maths|| Chapter 11 NCERT Notes

Circle:
A circle is a collection of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius (r).
Circumference of a Circle:
The perimeter or boundary of a circle is called its circumference.
- Formula: $Circumference = 2 π r$ where $r$ is the radius and $π \approx 3.14$ or $\frac{22}{7}$ .
Area of a Circle:
The area covered by a circle is calculated as:
- Formula: $Area = π r^{2}$

Areas of Sector and Segment of a Circle:

Sector of a Circle:
A sector is a portion of a circle enclosed by two radii and the corresponding arc. It resembles a "pizza slice" shape. The sector is characterized by the central angle ( $θ$ ) subtended by the arc at the center.
- Area of a Sector:
  The formula to calculate the area of a sector is derived from the proportion of the angle subtended at the center to the full angle (360°).
  $Area of Sector = (\frac{θ}{360}) \times π r^{2}$
  where $θ$ is the central angle in degrees and $r$ is the radius of the circle.
- Length of the Arc of a Sector:
  The length of the arc that subtends the angle $θ$ at the center is calculated as:
  $Arc Length = (\frac{θ}{360}) \times 2 π r$
  where $θ$ is the angle subtended and $r$ is the radius of the circle.
Segment of a Circle:
A segment is the region of a circle enclosed by a chord and the corresponding arc. There are two types of segments:
- Major Segment: The larger part of the circle.
- Minor Segment: The smaller part of the circle.
- Area of a Minor Segment:
  To calculate the area of a minor segment, we first find the area of the sector and subtract the area of the triangle formed by the two radii and the chord.
  $Area of Minor Segment = Area of Sector - Area of Triangle$
  The area of the triangle can be calculated using trigonometric formulas or the given dimensions.

Perimeter and Areas of Combinations of Plane Figures:

Area of a Circular Ring (Annulus):
A circular ring is the area between two concentric circles. The formula for the area of an annulus is given by the difference between the areas of the larger and smaller circles:
$Area of Circular Ring = π R^{2} - π r^{2} = π (R^{2} - r^{2})$
where $R$ and $r$ are the radii of the larger and smaller circles, respectively.
Areas of Combinations of Figures:
This involves calculating the areas of complex shapes that are a combination of circles, sectors, or other plane figures like squares and rectangles. Some common examples include:
- Finding the area of a circle inscribed in a square.
- Calculating the area of segments combined with triangles.
- Finding the area of sectors forming parts of polygons.

Important Formulas Recap:

Circumference of a Circle:
$Circumference = 2 π r$
Area of a Circle:
$Area = π r^{2}$
Area of a Sector of Angle $θ$ :
$Area of Sector = (\frac{θ}{360}) \times π r^{2}$
Length of an Arc of a Sector of Angle $θ$ :
$Arc Length = (\frac{θ}{360}) \times 2 π r$
Area of a Minor Segment:
$Area of Minor Segment = Area of Sector - Area of Triangle$
Area of Circular Ring (Annulus):
$Area of Circular Ring = π (R^{2} - r^{2})$

Example Problems:

Example 1 (Area of a Sector):
A sector of a circle has a radius of 7 cm and subtends an angle of 60° at the center. Calculate the area of the sector.
- Solution: $Area of Sector = (\frac{60}{360}) \times π \times 7^{2} = \frac{1}{6} \times π \times 49 = \frac{49 π}{6} \approx 25.67 {cm}^{2}$
Example 2 (Area of Segment):
A circle has a radius of 14 cm. A chord of the circle subtends an angle of 90° at the center. Calculate the area of the minor segment.
- Solution:
  First, calculate the area of the sector: $Area of Sector = (\frac{90}{360}) \times π \times 1 4^{2} = \frac{1}{4} \times π \times 196 = 154 {cm}^{2}$ Next, calculate the area of the triangle formed by the radii and the chord using trigonometric formulas or geometric properties.

Conclusion:

In Chapter 11: Areas Related to Circles, we learn important formulas for calculating areas and perimeters of circles, sectors, segments, and combinations of plane figures. Mastery of these concepts allows us to solve real-world problems involving circular shapes, from designing objects to calculating distances.

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