Arc Length Formula

Arc Length Formula – Arc length can be defined as the distance along a portion of the circumference of any circle or curve (arc). Arc length refers to any distance along the curved line that forms the arc. Arc is a segment of a curve or a portion of a circle’s circumference. They all have a curvature to their form. An arc’s length exceeds that of any straight line between its ends (a chord).

Arc Length Formula

The arc length formula is used to compute the distance along the curved line that forms the arc (a section of a circle). Arc length is simply the distance that passes through the curved line of the circle that forms the arc. It should be observed that the arc length exceeds the straight line distance between its ends.

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

Where,

  • s is the arc length
  • r is the radius of the circle
  • θ is the central angle of the arc

How to Find Arc Length of a Curve?

Based on the provided data, the arc length of a circle may be computed using several approaches and formulae. Some noteworthy cases are listed here.

Find the arc length using the radius and central angle.

The arc length of a circle can be calculated with the radius and central angle using the arc length formula,

  • Length of an Arc = θ × r, where θ is in radian.
  • Length of an Arc = θ × (π/180) × r, where θ is in degree.

Find the arc length without the radius and central angle.

The arc length of a circle can be calculated without the radius using:

Central angle and the sector area:

Central angle formulas

  • The sector area formula is, (θ/360º) × πr2, if θ is in degrees (or) (1/2) r2θ, if θ is in radians.
  • Use this formula and solve for the radius ‘r’. We need to use the square root in this process.
  • Then find the arc length using the relevant formula.

Example Questions Using the Formula for Arc Length

Question 1: Calculate the length of an arc if the radius of an arc is 10 cm and the central angle is 90°.

Solution:

Radius, r = 10 cm

Central angle, θ = 50°

Arc length = 2 π r ×  (θ/360°)

So, s = 2 × π × 10 × (90°/360°)

= 5 cm

 Practice Questions Based on Arc Length Formula

  1. What is the length of an arc produced by 75° of a circle with a diameter of 18 cm?
  2. An arc formed by 60° of a circle of radius “r” has a length of 8.37 cm. Calculate the radius (r) of that circle.
  3. Using the Arc Length Formula, calculate the perimeter of a semicircle of radius 1. cm.
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FAQs (Frequently Asked Questions)

The interspace between two locations along a portion of a curve is defined as the Arc Length Formula of a circle. A circle’s arc is any section of its circumference. The angle subtended by an arc at any point is the angle created by the two line segments connecting that point to the arc’s endpoints.

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

There are two equations associated with arc length. Given below are the two arc length equations.

  • Arc Length = θ × r, where θ is in radian.
  • Arc Length = rθ × (π/180), where θ is in degree

When arc length (L) is given with central angle θ then the circumference (C) is calculated using the equation L / C = θ/360º.