Arc Length Formula

Mathematics comes with a huge area of exploration for different types of studies related to calculations. Geometry is also a branch of maths that deals with angles, lines, segments, points, etc and helps us to determine the spatial relationship between them. It is one of the oldest parts of mathematics. As we there are different kinds or types of geometry that can be focused on. 

The given article is a study of arc length, arc, and formulas for determining arc length. The content provides different methods to calculate the length of an arc with examples. And, there are also some solved sample problems that give better clarification about the formulas and methods involved in calculating the length of an arc.

What is Arc Length?

Arc length is a curve or part of the circumference of a circle. It is the distance of the portion of a curve that forms an arc. All the arcs formed are curved in shape and can cover any distance along the direction of the curve. 

An arc can be defined as a part of a curve or circumference of a circle.

Arc length is part of the circumference or distance enclosed between two points on a curve. The two points forming the arc subtends an angle known as the central angle of the arc.

Arc Length Formula

The arc length formula is the formula used for the calculation of the length of an arc. The formulas to determine the length of the arc uses the central angle of the arc. These central angles are expressed in the forms of radians or degrees. The arc length of a circle is calculated by the product of θ times of the radius of the circle.

Mathematically the formula is written as 

In radian:

Arc length = θ × r

where,

θ is the angle expressed in radian

r is the radius of the circle

In degree:

Arc length = θ × (π/180) × r

Where,

θ is the central angle expressed in degree

r is the radius of the circle

How to find the arc length without a central angle?

Method 1: The arc length of the circle can be determined by using the radius and sector area of the circle in the condition where the central angle is unknown. The length of the arc without using the central angle can be determined by the given method.

• Step 1: Multiply the sector area of the given circle by 2.
• Step 2: Divide the number by the square of the radius. The central angle will be determined in this step.
• Step 3: Multiply the obtained central angle and the radius of the circle to get the arc length.

Example: Calculate the arc length of a curve sector area 50cm² and radius measuring as 4cm.

Solution:

Given

The sector area is 50cm²

The radius is 4cm.

Now,

=>sector area×2

=>50×2

=>100

And,

=>100/r²

=>100/4×4

=>6.25

6.25 is the central angle (In radian).

Then,

Arc length= radius×central angle 

=>4×6.25

=>25cm

Method 2: The arc length of the circle can be determined by using the radius and chord length of the circle in the condition where the central angle is unknown. The length of the arc without using the chord length and radius can be determined by the given method.

• Step 1: Divide the given chord length by twice the given radius.
• Step 2: Calculate the inverse of sine of the number obtained.
• Step 3: Multiply the result obtained by the inverse of sine of the number. In this step, the central angle is determined. The central angle obtained is expressed in radian.
• Step 4: Multiply the central angle by the radius to determine the arc length.

Example: Calculate the arc length of the curve which touches the chord of length 6cm and subtends a central angle of 2radians.

Solution:

Given

The central angle(θ) is 2 radian.

The chord length is 6cm.

Now, 

=>central angle/2

=>2/2=1

And, 

=>sin(1) = 0.841

And, 

=>chord length/(2×0.841)

=>6/1.682 = 3.56

This is the radius of the circle.

Then,

arc length = θ × r

=>arc length = 2×3.56=7.12cm

Sample Problems

Problem 1: Find the arc length of a curve on a circle with a radius of 4cm and central angle 2 radians.

Solution:

Given

The central angle(θ) is 2 radians.

The radius of the circle is 4cm.

Now,

Arc length=θ×r

=>2×4

=>8cm

Hence, the arc length is 8cm.

Problem 2: Find the arc length of a curve on a circle with a radius of 16cm and a central angle of 4radians.

Solution:

Given

The central angle(θ) is 4 radians.

The radius of the circle is 16cm.

Now,

Arc length=θ×r

=>4×16

=>64cm

Hence, the arc length is 8cm.

Problem 3: Calculate the arc length of a curve sector area 25cm² and radius measuring as 5cm.

Solution:

Given

The sector area is 25cm²

The radius is 5cm.

Now,

=>sector area×2

=>25×5

=>125

And,

=>125/r²

=>125/5×5

=>5

5 is the central angle (In radian).

Then,Arc length= radius×central angle

=>4×5

=>20cm

Problem 4: Calculate the arc length of a curve sector area 40cm² and radius measuring as 8cm.

Solution:

Given

The sector area is 40cm²

The radius is 8cm.

Now,

Arc length=sector area×2

=>40×8

=>320

And,

=>320/r²

=>320/8×8

=>5

5 is the central angle (In radian).

Then,

Arc length= radius×central angle

=>8×5

=>40cm

Problem 5: Calculate the arc length of the curve which touches the chord of length 16cm and subtends a central angle of 4radians.

Solution:

Given

The central angle(θ) is 2radian.

The chord length is 16cm.

Now,

=>central angle/2

=>2/2=1

And,

=>sin(1)=0.841

And,

=>chord length/(2×0.841)

=>16/1.682=9.51cm

This is the radius of the circle.

Then,

arc length =θ×r

=>arc length =2×9.51=19.02cm