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Area of a Sector


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The derivation of the area of a sector is presented

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Area of a Sector

  1. 1. Area of a Sector by SBR
  2. 2. Consider a circle with centre 𝑢 and radius 𝒓 units. Let 𝑨 and 𝑩 be any points on the circle. The area bounded by the arc 𝑨𝑩 and the two radii 𝑢𝑨 and 𝑢𝑩 is called the sector 𝑨𝑢𝑩. Let us find the area of the sector 𝑨𝑢𝑩.
  3. 3. Let the area of the sector be 𝑨. Let the angle subtended by the arc 𝑨𝑩 at the centre of the circle be 𝜽 (called angle of the sector). We know that in a circle, the area of the sector is proportional to the angle of the sector. Therefore, 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓 𝑨𝑢𝑩 βˆ π‘¨π‘Άπ‘© = 𝒂𝒓𝒆𝒂 𝒐𝒇 π’„π’Šπ’“π’„π’π’† πŸ‘πŸ”πŸŽ 𝒐 (but 360 π‘œ = 2πœ‹) β‡’ 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒆𝒄𝒕𝒐𝒓 𝑨𝑢𝑩 𝜽 = 𝒂𝒓𝒆𝒂 𝒐𝒇 π’„π’Šπ’“π’„π’π’† πŸπ…
  4. 4. β‡’ 𝑨 𝜽 = 𝝅𝒓 𝟐 πŸπ… β‡’ 𝑨 = 𝟏 𝟐 𝒓 𝟐 𝜽 Therefore, the area of the sector of radius 𝑨 = 𝟏 𝟐 𝒓 𝟐 𝜽