While studying the different parts of the circle, we read about the center, radius, diameter, secant, chord, segment, tangent, and the arc. In this article we will learn about the arc lengths of a circle, how to find its measure. We will also learn about the minor arc and the major arc.
What is an Arc of a Circle
An arc of a circle is a portion of the circumference of a circle bounded by two distinct points. More simply, it is a connected part of the circumference of a circle.
Shown below is an arc of the given circle.
Arc of a Circle
Types of Arcs of a Circle
Something quite interesting about the arc of a circle is that an arc is named according to the angle it subtends to the circle’s center. Accordingly, an arc is called a minor or a major arc.
Minor and Major Arcs
Major and Minor Arc of a Circle
The minor arc is an arc that subtends an angle of less than 180° to the center of the circle. In other words, the central angle of a minor arc measures less than a semicircle. In the given circle, AB is the minor arc. In contrast, a major arc is an arc that subtends an angle of more than 180° to the center of the circle. Thus, the central angle of a major arc measures more than a semicircle. In the given circle ACB is the major arc.
Formulas
As we know, the length of an arc is simply the distance covered by the curved line around the circle’s circumference. It depends on the radius of the circle and its central angle.
A circle is 360° all the way around. If we divide an arc’s degree measure by 360°, we will find the fraction of the circle’s circumference that the arc forms.
The relationship between circle’s circumference, arc length, and the central angle subtended by the arc is given as: Arc length = central angle/360° × circumference
How to Find the Arc Length of a Circle Given the Central Angle in Degrees
If the radius of the circle is known and the central angle subtended by the arc is given in degrees, the formula to find the arc length is given below:
Arc of a Circle Formula in Degrees
Let us work out some examples to understand the concept better.
An arc XY subtends an angle of 40 degrees to the center of a circle whose radius is 11 cm. Calculate the length of arc XY.
Solution:
As we know, Arc Length (L) = 2πr (θ/360), here θ = 40°, r = 11 cm, π = 3.141 = 2 × 3.141 × 11 (40/360) = 69.10/9 cm = 7.67 cm
Find the length of an arc of a circle having a radius of 6 m and that subtends an angle of 60 degrees to the center of the circle.
Solution:
As we know, Arc Length (L) = 2πr (θ/360), here θ = 60°, r = 6 m, π = 3.141 = 2 × 3.141 × 6 (60/360) = 6.28 m
How to Find the Arc Length of a Circle Given the Central Angle in Radians
The relationship between the angle subtended by an arc in radians and the ratio of the length of the arc to the radius of the circle is given as:
Central angle = Length of the arc/radius of the circle
Thus, if the radius of the circle is known and the central angle subtended by the arc is given in radians, the formula to find the arc length is given below:
Arc of a Circle Formula in Radians
The radian is thus the other way of measuring the size of an angle. For instance, to convert angles from degrees to radians, multiply the angle (in degrees) by π/180. Similarly, to convert radians to degrees, multiply the angle (in radians) by 180/π.
Let us work out some examples to understand the concept better.
Find the length of an arc whose radius is 20 cm and the angle subtended is 0.247 radians.
Solution:
As we know, Arc Length (L) = r × θ, here r = 20 cm, θ = 0.247 radians = (20 × 0.247) = 4.94 cm
Calculate the length of an arc which subtends an angle of 4.243 radians to the center of a circle which has a radius of 33 m.
Solution:
As we know, Arc Length (L) = r × θ, here r = 33 m, θ = 4.243radians = (33 × 4.243) m = 140.01 m
Find the angle subtended by an arc with a length of 11.05 mm and a radius of 4.5 mm.
Solution:
As we know, Arc Length (L) = r × θ, here r = 4.5 mm, θ = 11.05 mm = (4.5 × 11.05) mm = 49.72 mm
