Areas Related to Circles Class 10 Mathematics Concept notes and Formula Sheet

Concept Notes

1. Introduction to Circles

• A circle is a set of all points in a plane that are equidistant from a fixed point called the center.
• Important terms:
  • Radius (r): The distance from the center to any point on the circle.
  • Diameter (d): The longest distance across the circle, equal to twice the radius (d = 2r).
  • Circumference: The perimeter or boundary length of the circle.
  • Chord: A line segment that joins any two points on the circle.
  • Arc: A part of the circumference of the circle.
  • Sector: A region enclosed by two radii and the arc between them.
  • Segment: A region enclosed by a chord and the arc between them.

2. Area of a Circle

• The area enclosed by a circle is given by: $$\text{Area} = \pi r^2$$ where r is the radius of the circle and \pi (pi) is approximately 3.14159.

3. Circumference of a Circle

• The length of the boundary of the circle (circumference) is given by: $$\text{Circumference} = 2 \pi r$$ or equivalently, using the diameter, $$\text{Circumference} = \pi d$$

4. Area of a Sector

• A sector is a portion of a circle enclosed by two radii and the arc.
• The area of a sector with a central angle θ (in degrees) is given by: $$\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$$ If θ is in radians, the formula is: $$\text{Area of Sector} = \frac{1}{2} r^2 \theta$$

5. Length of an Arc

• The length of an arc corresponding to a central angle θ (in degrees) is given by: $$\text{Arc Length} = \frac{\theta}{360^\circ} \times 2 \pi r$$ If θ is in radians, the formula is: $$\text{Arc Length} = r \theta$$

6. Area of a Segment

• A segment is the region enclosed by a chord and the arc between the chord's endpoints.
• The area of a segment is calculated as: $$\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle formed by the chord and the radii}$$

7. Area of a Triangle in a Circle

• When the circle's center is at the origin, the area of a triangle formed by two radii and the chord can be calculated using trigonometry or by using the formula: $$\text{Area of Triangle} = \frac{1}{2} \times r^2 \times \sin(\theta)$$ where θ is the central angle in radians.

8. Important Theorems and Properties

• Tangent-Secant Theorem: The square of the length of a tangent segment from a point outside the circle is equal to the product of the lengths of the entire secant and its external segment.
• Area of Annulus: The region between two concentric circles is called an annulus. The area is given by: $$\text{Area of Annulus} = \pi (R^2 - r^2)$$ where R is the radius of the larger circle and r is the radius of the smaller circle.

Formula Sheet

1. Area of a Circle:

   $$\text{Area} = \pi r^2$$

2. Circumference of a Circle:

   $$\text{Circumference} = 2 \pi r \quad \text{or} \quad \pi d$$

3. Area of a Sector (θ in degrees):

   $$\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$$

4. Area of a Sector (θ in radians):

   $$\text{Area of Sector} = \frac{1}{2} r^2 \theta$$

5. Length of an Arc (θ in degrees):

   $$\text{Arc Length} = \frac{\theta}{360^\circ} \times 2 \pi r$$

6. Length of an Arc (θ in radians):

   $$\text{Arc Length} = r \theta$$

7. Area of a Segment:

   $$\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}$$

8. Area of Annulus:

   $$\text{Area of Annulus} = \pi (R^2 - r^2)$$

9. Area of a Triangle (formed by radii and chord):

   $$\text{Area of Triangle} = \frac{1}{2} r^2 \sin(\theta)$$

Examples for Practice

1. Find the area of a sector with a central angle of 60° in a circle of radius 10 cm.
2. Calculate the length of an arc subtended by a central angle of 120° in a circle with a radius of 14 cm.
3. A chord of a circle of radius 15 cm subtends a central angle of 90°. Find the area of the segment formed.