What Is The Arc Length Formula? How To Find The Length Of An Arc Using The Corre

Posted by Rosalie Galvez on December 21st, 2020

Arc is a portion of circumference or the curved path with a constant radius stretching from point A to point B. Arc length is the measure of the length of this arc. Arc have a central point around which it is curved, it is the center of the circle that would be formed if the arc is given a full 360o rotation. The other characteristic it has is the angle between the started and ending points if a line was stretched between the end points of arc and the center point, then the angle between these two lines will be the central angle.

All of these terminologies may be getting overwhelming but just take a breath, and look at the figure illustrated below, this will clear all the doubts and concepts about arcs.

Now that we know what an arc is we can move down to calculating the arc length. There is a simple relation the related the arc length, radius of the arc and the central angle. The relation is

s = r × ?

Here ‘s’ is the arc length

‘r’ is the radius of the arc

‘?’ is the central angle given in radians

Now we see that the arc length is simply the product of the central angle and radius of curvature.

To fully understand this formula, lets work out an example and see this relation is action

Lets say we are tasked to find the length of an arc which draws a central angle of 45o and has a radius of 50 cm.

First step is to write down all the data we are given

r = 50 cm

? = 45°

Remember! We are supposed to convert degree into radian before proceeding further

To convert into radian we multiply it with ?/180

? =45×?/180

? = ?/4 rad

Now to calculate the arc length we take the product of both these quantities

s=r×?

s=50×?/4

s=25?/2

Or

s=39.27 cm

Alternatively if we are given the arc length and either the radius or the central angle, we can find out the remaining variables by simply rearranging the equation to get the required variables.

Fun fact: We can also calculate the area of the sectors. Sectors are portions of the circle which any arc length separates, a sector is pretty much a slice of a pizza from the whole round pizza. Now we can calculate its area and see how much pizza you’re eating.

Arc length formula can be pretty useful when finding out how much distance does a wheel travel if it is given a certain amount of rotations. We can use this formula to convert rotational motion into linear translatory motion. It can also be pretty helpful in finding out how many inches of travel in pulley would lift the certain amount of weight. This simple formula will go a long way of you are studying calculus and trigonometry, so you better grab onto it tight!

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