📚 Concepts

📚 Circle and Its Elements

• A circle is a closed curved shape with all points at the same distance from a central point.

📚 Radius and Diameter

• The radius is the distance from the center to any point on the circle. The diameter is twice the radius and passes through the center.

📚 Circumference

• The total distance around the circle is called the circumference.

📚 Area of a Circle

• The total space enclosed within the boundary of a circle.

📚 Sector of a Circle

• A sector is a portion of a circle enclosed between two radii and the connecting arc.

📚 Segment of a Circle

• A segment is the region between a chord and the arc lying between the chord's endpoints.

📚 Minor and Major Arcs

• A minor arc is smaller than a semicircle, while a major arc is larger than a semicircle.

📚 Minor and Major Sectors

• A minor sector has a central angle less than 180°, while a major sector has more than 180°.

📚 Minor and Major Segments

• A minor segment is smaller and lies below the chord. A major segment is larger and lies above it.

✍️ Formulas

✍️ Circumference of a Circle

• 𝐶 = 2 × π × 𝑟 (Where 𝑟 is the radius of the circle)

✍️ Area of a Circle

• 𝐴 = π × 𝑟²

✍️ Length of an Arc (Central angle = θ°)

• 𝐿 = (θ⁄360) × 2 × π × 𝑟

✍️ Area of a Sector (Central angle = θ°)

• 𝐴ₛ = (θ⁄360) × π × 𝑟²

✍️ Perimeter of a Sector

• 𝑃 = (θ⁄360) × 2 × π × 𝑟 + 2𝑟 (Arc length + 2 radii)

✍️ Area of a Segment (Minor segment)

• 𝐴ₛₑ = 𝐴ₛ − 𝐴ₜ = (θ⁄360) × π × 𝑟² − ½ × 𝑟² × sinθ

  (Where θ is in degrees and 𝐴ₜ is the area of triangle formed by two radii)

✍️ Area of Major Sector or Segment

• Area of Major Sector = π × 𝑟² − Area of Minor Sector

• Area of Major Segment = π × 𝑟² − Area of Minor Segment

✍️ Area of Triangle (used in Segment formula)

• If θ is the central angle in radians or degrees:  

  𝐴ₜ = ½ × 𝑟² × sinθ

• For triangle with known base and height:  

  𝐴ₜ = ½ × base × height

✍️ Area of a Ring (Annular Region)

• 𝐴ₐ = π × (𝑅² − 𝑟²)

  (Where 𝑅 is outer radius and 𝑟 is inner radius)

  
  

✍️ Total Area Involving Circle + Other Shapes

• Circle Inside Square:

  • Shaded Area = Area of Square − Area of Circle = (side)² − π × 𝑟²

• Semi-circle on one side of square: 

  • Total Area = Area of Square + (½ × π × 𝑟²)

• Quarter Circle:  

  • Area = (¼) × π × 𝑟²  

  • Perimeter = (¼ × 2πr) + 2r = (½ × π × 𝑟) + 2𝑟

• Semi-circle:  

  • Area = (½) × π × 𝑟²  

  • Perimeter = π𝑟 + 2𝑟

✍️ Shaded Region in Compound Figures

• Area of shaded part = Area of larger shape − Area of inner/removed circle/sector/triangle as applicable

  (This applies in diagrams with combinations like triangle + sector, square − quarter circle, etc.)