Understand the Concept of Arc Length Formula & How to Find Arc Length of a Circle?
In this article, you will be able to understand what arc length is. You will also learn how to calculate arc length between two points.
What is Arc Length?
The arc length is defined as the distance covered by a curve between two points. It helps to determine the length of an irregular or curved shape. In other words, it is the measure of length of a curved path. Sometimes, the arc length of a circle may be a part of its circumference.
It is useful to calculate arc length of a curve in space because it determines the length of an arc of the curve so that you may be able to calculate the length of a complete curve. It has many applications in geometry, for example, it helps to calculate the circumference of a circle.
Arc Length Formula
The arc length can be calculated by different formulas based on the unit of the central angle of the arc. It is measured in degrees or radians. The formula of arc length is:
There is also a different formula to measure arc length in radian that is:
$$ L\;=\; θ \;×\; r $$
To calculate arc length, the radius and central angle of the circle should be known.
Derivation of Arc Length Formula
The circumference of a circle is also itself an arc. So the angle made by the circumference at the centre of the circle is 360 degrees.
The length of circumference is,
$$ L\;=\; 2\pi r $$
So the angle made by an arc with length C at the centre will be,
$$ θ \;=\; \frac{360^∘}{2\pi r}×L $$
i.e.
$$ θ \;=\; \frac {180^\circ}{\pi r}×L $$
And
$$ L\;=\; \frac{θ}{180^\circ}×\pi r $$
Which is the formula to find the arc length of an arc.
How to Find Arc length?
The arc length of an arc of a circle can be calculated by using the given data. There are two cases to calculate arc length based on given values of radius and central angle. These cases are:
- Arc length in degrees
- Arc length in radian
See the examples below to understand how to calculate arc length.
FAQ’s
What is Meant by Arc Length?
The arc length is referred to as the length of a line segment of a curve. It means the distance between two points on a curve.
What is the Arc Formula?
The formula to measure arc length of an arc of a circle in degrees:
$$ L \;=\; 2 \pi r × \frac{θ}{360} $$
And the formula to calculate arc length in radian is:
$$ L\;=\; θ × r $$
Where,
r = radius of the circle,
θ = central angle in degrees
