Arc Length
A part of a curve or a part of a circumference of a circle is called Arc. All of them have a curve in their shape. The curved portion of these objects is mathematically called an arc. Arc length is better defined as the distance along the part of the circumference of any circle or any curve (arc). Any distance along the curved line that makes up the arc is known as the arc length. The length of an arc is longer than any straight line distance between its endpoints (a chord).
1.  What is Arc Length? 
2.  Arc Length Formula 
3.  How to Find Arc Length of a Curve? 
4.  FAQs on Arc Length 
What is Arc Length?
The arc length is defined as the interspace between the two points along a section of a curve. An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the endpoints of the arc. For example, in the circle shown below, OP is the arc of the circle with center Q. The arc length of this arc OP is given as L.
Arc Length Formula
The length of an arc can be calculated using different formulas, based on the unit of the central angle of the arc. The measurements of the central angle can be given in degrees or radians, and accordingly, we calculate the arc length of a circle. For a circle, the arc length formula is θ times the radius of a circle.
The arc length formula in radians can be expressed as, arc length = θ × r, when θ is in radian. Arc Length = θ × (π/180) × r, where θ is in degree, where,
 L = Length of an Arc
 θ = Central angle of Arc
 r = Radius of the circle
Arc Length Formula in Radians
The arc length of a circle can be calculated using different formulas, based on the unit of the center angle of the arc. The arc length formula in radians can be expressed as,
Arc Length = θ × r
where,
 L = Arc Length
 θ = Center angle of the arc in radians
 r = Radius of the circle
How to Find Arc Length of a Curve?
The arc length of an arc of a circle can be calculated using different methods and formulas based on the given data. Some important cases are given below,
 find arc length with the radius and central angle
 find arc length without the radius
 find arc length without the central angle
How to Find Arc Length With 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.
How to Find Arc Length Without the Radius?
The arc length of a circle can be calculated without the radius using:
Central angle and the sector area:
 Multiply the sector area by 2 and further, divide the result by the central angle in radians.
 Find the square root of the result of the division.
 Multiply this obtained root by the central angle again to get the arc length.
 The units of this calculated arc length will be the square root of the sector area units.
Example: Calculate the arc length of a curve with sector area 25 square units and the central angle as 2 radians.
We have,
Sector area = 25 units
Central angle = 2 radians
 Step 1: Sector area × 2 = 25 × 2 = 50
 Step 2: 50/central angle = 50/2 = 25
 Step 3: √25 = 5
 Step 4: 5 × central angle = 5 × 2 = 10 units
Thus, arc length = 10 units
Central angle and the chord length:
 Divide the central angle in radians by 2 and further, perform the sine function on it.
 Divide the given chord length by twice the result of step 1. This calculation gives you the radius as result.
 Multiply the radius by the central angle to get the arc length.
Example: Calculate the arc length of a curve, whose endpoints touch a chord of the circle measuring 5 units. The central angle subtended by the arc is 2 radians.
We have,
Chord length = 5 units
Central angle = 2 radians
 Step 1: Central angle/2 = 2/2 = 1
 Step 2: Sin(1) = 0.841
 Step 3: Chord length/ (2 × 0.841) = 5/ 1.682 = 2.973 units = radius
 Step 4: Arc length = radius × central angle = 2.973 × 2 = 5.946 units
Thus, arc length = 5.946 units
How to Find Arc Length Without the Central Angle?
The arc length of a circle can be calculated without the angle using:
Radius and the sector area:
 Multiply the sector area by 2.
 Then divide the result by the radius squared (the units should be the same) to get the central angle in radians.
 Multiply the central angle by the radius to get the arc length.
Example: Calculate the arc length of a curve with sector area 25 square units and radius as 2 units.
We have,
Sector area = 25 units
Central angle = 2 units
 Step 1: Sector area × 2 = 25 × 2 = 50
 Step 2: 50/radius^{2} = 50/4 = 12.5 = central angle(rad)
 Step 3: Arc length = radius × central angle = 2 × 12.5 = 25 units
Thus, arc length = 25 units
Radius and chord length:
 Divide the chord length by twice the given radius.
 Find the inverse sine of the obtained result.
 Double the result of the inverse sine to get the central angle in radians.
 Multiply the central angle by the radius to get the arc length.
Example: Calculate the arc length of a curve, whose endpoints touch a chord of the circle measuring 5 units. The radius of the circle is 2 units.
We have,
Chord length = 5 units
Central angle = 2 units
 Step 1: Chord length/(2 × radius) = 5/(2 × 2) = 1.25
 Step 2: Sin^{1}(1.25) = 0.949
 Step 3: Central angle = 2 × 0.949 = 1.898 radians
 Step 4: Arc length = radius × central angle = 2 × 1.898 = 3.796 units
Thus, arc length = 3.796 units
Important Notes
Given below are key highlights on the concept of arc length.
 Arc Length = θ × r, where θ is in radian.
 Arc Length = θ × (π/180) × r, where θ is in degree.
Related Topics on Arc Length:
 Circles
 Central Angle
 What is a Radian?
 Angles Formulas
 Radius
 Circumference of a circle
 Area of a Circle
 Diameter
 All Circle Formulas
Solved Examples on Arc length

Example 1: Find the length of an arc cut off by a central angle of 4 radians in a circle with a radius of 6 inches.
Solution:
Center angle, θ = 4 radians, radius, r = 6 inches . Use the arc length formula, L = θ × r = 4 × 6 = 24 inches. ∴ Arc length (PQ) = 24 inches

Example 2: Find the length of an arc cut off by a central angle, θ = 40º in a circle with a radius of 4 inches.
Solution:
Radius, r = 4 inches , θ = 40º. Use the arc length formula, L = π × (r) × (θ/180º) = π × (4) × (40º/180º) = 2.79 inches. ∴ Arc length (P0) = 2.79 inches
FAQs on Arc Length
What is Arc Length of a Circle?
The arc length of a circle is defined as the interspace between the two points along a section of a curve. An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the endpoints of the arc.
How do you Find the Length of an Arc Without the Radius?
The arc length of a circle can be calculated without the radius using:
Central angle and the sector area:
 Multiply the sector area by 2 and further, divide the result by the central angle in radians.
 Find the square root of the result of the division.
 Multiply this obtained root by the central angle again to get the arc length.
 The units of this calculated arc length will be the square root of the sector area units.
Central angle and the chord length:
 Divide the central angle in radians by 2 and further, perform the sine function on it.
 Divide the given chord length by twice the result of step 1. This calculation gives you the radius as result.
 Multiply the radius by the central angle to get the arc length.
How do you Find Arc Length Using Radians?
The arc length can be calculated when the central angle is given in radians using the formula: Arc Length = θ × r, when θ is in radian.
 L = Arc Length
 θ = Center angle of the arc
 r = Radius of the circle
Does Arc Length have to be in Radians?
No, arc length cannot be in radians. It is a measurement of distance, so cannot be in radians. The central angle subtended at the center can be in radians, degrees, or arcsecs accordingly.
How do you Find the Circumference of Arc Length?
When arc length is given with central angle θ then the circumference is calculated as Arc Length (L)/Circumference = θ/360º.
What is the Length of Major Arc Using Arc Length Formula?
A major arc in a circle is larger than a semicircle. It is measured as larger than 180°. Using formula ℓ = rθ we can find the length of an arc of a circle, where θ is in radian.