How to find radius from arc length

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. 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 length of an arc is longer than any straight line distance between its endpoints (a chord).

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 end-points 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/radius2 = 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 on Arc Length

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

Check out a few more interesting articles related to arc length to understand the topic more precisely.

1. 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

2. Question 2: The radius of the circle is 14 units and the arc subtends 65° at the center. What is the length of the arc?

   Solution: We know that,

   Circumference of circle = 2πr

   C = 2π × 14 = 28π

   arc length = (θ/360) × C = (65°/360°)28π = 15.882 units

3. Example 3: 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

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FAQs on Arc Length

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 end-points 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.

What do You Understand By Arc Lenght Equation?

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 = θ × (π/180) × r, where θ is in degree

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.