Arc Length Formula

Arc Length Formula

The Arc Length Formula is more accurately described as the distance along a portion of the circumference of any circle or curve (arc). The Arc Length Formula is any distance along the curved line that forms the arc. Arc is a segment of a curve or a segment of a circle’s circumference. They are all shaped with a curve. An arc’s length is greater than any straight line distance between its endpoints (a chord).

Formulas for Arc Length

An arc is any linked portion of a circle’s circumference. An arc length is a distance between two points on an arc. To calculate an arc length, we must first understand the geometry of a circle.

Because the arc is a fraction of the circle’s circumference, and we know that the entire circumference makes a 360-degree angle to the circle’s centre. So, given the circumference part, i.e. arc, and the central angle of the arc, we can simply calculate the length of the arc.

An angle is a geometrical figure formed when two lines intersect at the same place on a plane. The length of an arc in a given circle is a fraction of its circumference. The Arc Length Formula is the distance along the curved line that makes up the arc. It will be longer than the chord, which is the straight line distance between its endpoints.

The length of an arc may be computed using several formulae dependent on the unit of the arc’s central angle. The centre angle can be measured in degrees or radians, and the Arc Length Formula of a circle is calculated appropriately. The Arc Length Formula for θ a circle is times the radius of a circle.

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

Denotations in the Arc Length Formula

The denotations of the Arc Length Formula are as follows

The arc length is s

The radius of the circle is r

The central angle of the arc is θ

Example Questions Using the Formula for Arc Length

  1. Determine the length of an arc if its radius is 8 cm and its central angle is 40°.

Central angle, θ = 40°

The Arc Length Formula = 2 π r × (360°)

So, s = 2 × π × 8 × (40°360°)

= 5.582 cm 

  1. What is the length of the arc for the function f(x) = 6 between x = 4 and x = 6?

Because the function is a constant, its differential will be 0. As a result, the arc length will now be-


So, arc length s = (6 – 4) = 2. 

  1. What is the Arc Length Formula of a Circle?

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°) 

  1. How do One Calculate the Length of an Arc without knowing the Radius?

The arc length of a circle may be computed without knowing the radius by using the formula:

 Central angle and the sector area:

Divide the result by the central angle in radians after multiplying the sector area by two.

Determine the square root of the division result.

To acquire the arc length, multiply the obtained root by the central angle once again.

The square root of the sector area units will be used to compute the arc length.

The chord length and the central angle: 

Divide the centre angle in radians by two and apply the sine function to it.

Divide the result of step 1 by the provided chord length. The radius is the outcome of this computation.

To get the arc length, multiply the radius by the centre angle. 

 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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