How to Find Arc Length and Arc Length of a Curve?

Formula for Arc Length

Arc Length

Arc length is more accurately described as the length along the perimeter of any circle or curve (arc). The arc length is the measurement of any distance along the curved path that forms the arc. Arc refers to a segment of a curve or the circumference of a circle. Each of them is shaped like a curve. An arc’s length is more than the distance in a straight line between its endpoints (a chord).

 

 Arc Length Definition

 

The distance between two places in a curve’s section is known as the arc length. Any portion of a circle’s circumference is an arc. An arc’s Arc at any given location is the Arc created by the two line segments connecting that point to the arc’s endpoints. As in the circle below, where Q is the center, OP is the arc of the circle. This arc OP’s arc length is indicated by the letter L.

 

Formula for Arc Length

Depending on the unit of the arc’s central Arc, many formulas can be used to determine the length of an arc. We determine the arc length of a circle based on the measurements of the center Arc, which can be expressed in degrees or radians. The arc length formula for a circle is times the radius of a circle.

You can calculate the arc length of a circle easily within seconds by using arc length integral calculator. This tool follow the same formula and provides accurate results according to input values.

 

When r is in radians, the formula for arc length can be written as arc length = r. Arc Length = (/180) r, where r is the radius, and where

 

L stands for length and for the central Arc of the arc.

 

r = Circle’s radius

 

Radians Formula for Arc Length

 

Depending on the unit of the arc’s center Arc, various formulas can be used to determine the arc length

Of a circle. Arc Length = r where, is the radian equivalent of the arc length formula.

 

L = Arc Length

Arc’s Center Arc in Radians

r = Circle’s radius

 

How to Calculate a Curve’s Arc Length?

Various techniques and formulas can be used to determine the arc length of a circle’s arc based on the provided information. You can find the distance between two points on the curve line in polar coordinates more efficient way with the help of arc length of polar curve. Below, some significant cases are provided.

 

  • Using the radius and central Arc, find the arc length.
  • Find the length of the arc without the radius and the center Arc.

How Can I Calculate Arc Length Using Radius and Central Arc?

Using the arc length formula, one may determine a circle’s arc length given its radius and center Arc.

  • An arc’s length is equal to r, where r is the radian unit.
  • The length of an arc is equal to (r/180), where r is in degrees.

 

How Can I Calculate Arc Length Without a Radius?

Without knowing the radius, one can determine the arc length of a circle by using:

The sector area and the central Arc:

 

  • Divide the result by the central Arc in radians after multiplying the sector area by two.
  • Calculate the square root of the division outcome.
  • To determine the arc length, multiply the newly found root by the center Arc.
  • The square root of the sector area units will be used as the units for this estimated arc length.
  • Calculate the length of the arc for a curve with a sector size of 25 square units and a central Arc of 2 radians, for instance.

 

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We’ve got

 

  • Area sector = 25 units
  • 2 radians is the central Arc.
  • Sector area 2 = 25 2 = 50 in
  • 50/center Arc = 50/2 = 25
  • √25 = 5
  • 5 central Arcs equals 5 times two, or 10 units.
  • So, arc length equals 10 units.

 

The chord length and the central Arc:

Multiply the central Arc in radians by two before applying the sine function.

Divide the result of step 1 by twice the chord length that is specified. The radius is the output of this calculation.

To calculate the arc length, multiply the radius by the central Arc.Calculate the arc length of a curve whose endpoints touch a circle’s 5-unit chord, for instance. The arc’s center is subtended by an Arc of 2 radians.

 

The chord length is five units.

 

  • 2 radians is the central Arc.
  • Central Arc/2 = 2/2 = 1
  • Sin(1) = 0.841
  • Radius = Chord length / (2 0.841) / 5 / 1.682
  • 946 units (arc length = radius central Arc)
  • Arc length therefore equals 5.946 units.

 

How Can I Calculate Arc Length Without Knowing the Central Arc?

 

Without knowing the Arc, one can determine the arc length of a circle by using:

The sector area’s radius:

  • Divide by two the sector area.
  • To obtain the central Arc in radians, divide the result by the radius squared (the units should be the same).

 

To calculate the arc length, multiply the central Arc by the radius.

 

Calculate the length of the arc for a curve with a sector area of 25 square units and a radius of 2 units, for instance.We have 25 units in the sector.

 

Central Arc equals two units.

 

  • Sector area 2 = 25 2 = 50
  • 50/radius2 = 50/4 = 12.5 = the central Arc (rad)
  • Arc length = radius central Arc = 2 12.5 = 25 units
  • So, arc length equals 25 units.
  • The chord’s length and radius.
  • Multiply the radius by twice the chord length.
  • Calculate the inverse sine of the outcome.
  • To find the central Arc in radians, multiply the inverse sine result by two.
  • To calculate the arc length, multiply the central Arc by the radius.

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Example:

Determine the length of the arc formed by a curve whose ends touch a five-unit chord of the circle. The circle has a radius of two units.

The chord length is five units.

  • Central Arc equals two units.
  • 5/(2 2) = 1.25 for the chord length/(2 radius) formula.
  • Sin-1(1.25) = 0.949
  • The central Arc is equal to 2 x 0.949, or 1.898 radians.
  • Divide the arc’s length by its radius and central Arc to get 3.796 units.
  • Arc length therefore equals 3.796 units.
  • Relevant Information on Arc Length
  • The key points on the idea of arc length are listed below.
  • Arc Length is equal to r, where r is the radian unit.
  • Arc Length is equal to r (/180), where is a degree.

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