The document discusses how to calculate the length of an arc on a circle. It explains that the ratio of any arc length to the angle it subtends at the center is equal to the ratio of the radius to a full circle. Therefore, the length of any arc is equal to the radius multiplied by the central angle subtended by the arc in radians.

1. Arc length
by
SBR
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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.

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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.