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Tools City

Arc Length Calculator

Tools City's Arc Length calculator is a tool that can calculate the length of an arc, the sector area of a circle, and the chord length. This article explains the arc length formula in detail and provides you with step-by-step instructions on finding the arc length, chord length, and sector area of a circle.

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What do you mean by Arc Length?

If there are two points along the section of the curve the distance between them is known as the length of an Arc or an Arc length.

In other words, the distance that goes through a curved line of a circle creating the arc is said to be the Arc length or the length of an Arc. It is noted that the arc length is more extended than the straight line distance between its endpoints.

What is an Arc Length Formula?

We will use the Arc length formula to evaluate the distance along the curved line making up the arc, a circle segment.

arc length

For any circle:

If the θ is in radians

Arc Length = θ × r

If the θ is in degrees

Arc Length = 2 π r ×  (θ / 360°)

 


The length of an arc relies on two things; the radius of a circle and the central angle denoted with 'θ' theta. We know that we can calculate angles in degrees and radians, so 360 degrees is always equal to 2π radians. The arc length of 360 degrees and 2π radians is similar to its circumference. Therefore, as the proportion between angle and arc length is constant, we can conclude that:


L / θ = C / 2 π


As for circumference C = 2 π r, now substituting the value of C in the above equation


L / θ = 2 π r / 2 π

L / θ = r


We find out the arc length formula in radians when multiplying this equation by θ:


L = r * θ


Therefore, the arc length (in radians) is equal to the product of radius and the central angle.

Example Question Using the Formula for Arc Length

Question: Calculate the length of an arc if the radius of an arc is 4 cm and the central angle is 60°.


Data: 

Radius of an arc = r = 4 cm

Central angle = θ = 60°

Length of an arc = L = ?


Solution:

As we already know, the formula for the length of arc when the central angle is given in degrees;

Arc length = 2 π r × (θ/360°)

Arc length = 2 × π × 4 × (60°/360°)

Arc length = 4.189 cm

How you can figure out the Area of a sector of a circle?

Similarly, we can figure out the area of a circle's sector. The area of any circle is equal to π r². Therefore,


A / θ = πr² / 2π

A / θ = r² / 2


The formula for the area of a sector will be:


A = r² * θ / 2

 

Example Question Using the Formula for Area of a sector of a circle

Question: What is the Area of the circle's sector if the circle radius is 7 millimetres and the angle of the sector is 40 radians?

 

Data

Radius of a circle = r = 7 mm

Central angle = θ = 40 rad

Area of the sector = A = ?

 

Solution:

As we already know, the formula for the Area of a sector:

 

Area of sector = r² * θ / 2

Area of sector = 7² * 40 / 2

Area of sector = 49 * 40 / 2

 

Area of sector = 980 mm²

 

How to calculate the Chord of a Circle? 

The chord of a circle can be defined as a line segment joining two points on the circle's circumference. The diameter is the most extended chord of the circle that passes through the circle's center.

There are two basic formulas for finding the length of the chord of a circle are as follows:

  1. Chord Length Using Perpendicular Distance from the Centre of the circle:
    Chord length = 2 square root r²-d²

  2. Chord Length Using Trigonometry with angle θ:
    Chord length = 2 × r × sin (θ/2)

Example Question Using the Formula for Chord of a circle

Question: Calculate the length of the chord where the radius of the circle is 7 mm, and the perpendicular distance drawn from the center of the circle to its chord is 4 mm.

 

Data: 

Radius of a circle = r = 7 mm

Perpendicular distance = d = 4 mm

Chord length = c = ?

 

Solution:

As we already know, the formula for the chord length.

 

Chord length = 2 square root r²-d²

 

Chord length = 2 square root 7²-4²

 

Chord Length = 2 square root 33

 

Chord Length = 11.488 mm

Which tool is best for calculating the Arc length of a circle?

Tools City’s Arc Length calculator is one of the best tools you will see all over the internet. It calculates not only the arc length of a circle but also measures the chord length, sector area, and much more. You can use many different units in Tools City’s Arc Length calculator as it provides a variety of units used in daily life, i.e, metre, centimetre, kilometres, etc.

How to use Tools City’s Arc Length calculator?

Tools City's Arc Length calculator is very easy to use; you must know the values of a few parameters. If you want to calculate Arc length, you must be aware of the radius 'r' and central angle 'θ'. Following are the few steps to use Arc Length calculator.

  1. Log on to the Tools City's Arc Length calculator.

  2. You will see a picture and six different boxes for different values. The first box is for central angle, the second box for radius, the third for diametre, the fourth for sector area, the fifth for chord length, and the final for Arc Length. 

  3. You will now put the values for the parameters you know. For example, you put 1 mm in the radius box and 45 degrees in the central angle box. You will come up with all those remaining boxes as its results.