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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
  2. 2. Consider a circle with centre 𝑶 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 ∠𝑨𝑶𝑩 = 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. We know that in a circle, the arc lengths are proportional to the angles subtended by them at the centre. Therefore, 𝒂𝒓𝒄 𝑨𝑩 ∠𝑨𝑶𝑩 = 𝒂𝒓𝒄 𝑷𝑨 ∠𝑷𝑶𝑨 ⇒ 𝒓 𝟏 𝒄 = 𝒔 𝜽 𝒄 ⇒ 𝜽 𝒄 𝟏 𝒄 𝒓 = 𝒔 ⇒ 𝒔 = 𝒓𝜽 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.