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Arc length

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Derivation of formula to find the length of an arc given the radius of the arc and the angle subtended at the center of the arc.

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Arc length

  1. 1. Arc length by SBR www.harekrishnahub.com
  2. 2. www.harekrishnahub.com Consider a circle with centre 𝑢 and radius 𝒓 units. Let 𝑨 and 𝑩 be any points on the circle, such that the length of the arc 𝑨𝑩 is equal to the radius 𝒓 of the circle. We have βˆ π‘¨π‘Άπ‘© = 𝟏 𝒄 . Let 𝑷 be any point on the circle. Let the arc 𝑷𝑨 subtend an angle 𝜽 𝒄 at the centre of the circle. Let 𝒔 be the length of the arc 𝑷𝑨. Let us find the length of the arc 𝑷𝑨.
  3. 3. www.harekrishnahub.com We know that in a circle, the arc lengths are proportional to the angles subtended by them at the centre. Therefore, 𝒂𝒓𝒄 𝑨𝑩 βˆ π‘¨π‘Άπ‘© = 𝒂𝒓𝒄 𝑷𝑨 βˆ π‘·π‘Άπ‘¨ β‡’ 𝒓 𝟏 𝒄 = 𝒔 𝜽 𝒄 β‡’ 𝜽 𝒄 𝟏 𝒄 𝒓 = 𝒔 β‡’ 𝒔 = π’“πœ½ Therefore, the length of the arc is given by the product of the radius of the arc and the angle (in radians) subtended by the arc at the centre.

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