What is Arc Length?

Definition: Arc length is the distance between two points in a curve. It differs from a straight line because curves have unequal slopes and hence arc lengths would require additional computation.

The arc length formula is calculated to measure the distance along the curved line making up the arc, which is just a portion of a circle. In simple terms, it is referred to as the arc length, understood to be the distance running through the curved line of the circle that makes up the arc. It is also important to understand that the arc length is longer than the distance between its endpoints.

Formulas for Arc Length

The formula to compute the length of the arc is,

Arc Length Formula (if θ is in degrees):   s = 2 π r (θ/360°)

Arc Length Formula (if θ is in radians):     s = ϴ × r

Arc Length Formula in Integral Form:       s= 

Denotations in the Arc Length Formula

• s is the arc length

• r is the radius of the circle

• θ is the central angle of the arc

Example Questions Using the Formula for Arc Length

Question 1: Calculate the length of an arc if the radius of an arc is 8 cm and the central angle is 40°.

Solution:

Radius, r = 8 cm

Central angle, θ = 40°

Arc length = 2 π r × (θ/360°)

Hence, s = 2 × π × 8 × (40°/360°)

= 5.582 cm

Question 2: Find the length of an arc formed by a function f(x) = 6 in the interval between x = 4 and x = 6.

Solution:

Since the function is a constant, so its differential will be 0. Hence, the arc length now will be

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