How to Find Length of Arc of a Circle?

Here’s a step-by-step explanation of how to find the length of an arc of a circle, using an example.

Assume we have a circle with a radius of 10 units and a centre angle of 120°. The length of the arc subtended by this angle is what we’re looking for.

Step 1: Check the Given values

• Radius (r) = 10 units
• Central angle (θ) = 120°

Step 2: Calculate Arc Length

Using the formula for arc length:
Arc Length = (θ/360°) × 2πr

Step 2.1: Calculate Fraction of Circle’s Circumference

θ / 360° = 120°/360° = 1/3

Step 2.2: Calculate Arc Length

Arc Length = (1/3) × 2π × 10

⇒ Arc Length = (2/3)π × 10

⇒ Arc Length = (20/3)π units

Step 3: Finalize the Result

If you want a numerical estimate, the value of π (pi) is about 3.14.

Thus, Arc Length ≈ (20/3) × 3.14 ≈ 20.93 units

The arc subtended by a central angle of 120° in a circle with a radius of 10 units is roughly 20.93 units long.

How to Find the Arc Length in Radians?

The angle that an arc occupies in radians and the proportion of the arc’s length to the circle’s radius is related. In this instance.

θ = (Length of an Arc)/(Radius of the Circle)

OR

S = r θ

Where,

• θ is the angle in radians that an arc occupies,
• S is the angle’s length, and
• r is the radius of the given circle.

Note:

• For θ = 1 radian, or s = r, is the center angle that a radius-length arc subtends.
• The radian is merely another unit of measurement for angles. For instance, multiply the angle (in degrees) by π/180 to convert angles from degrees to radians.
• The angle (in radians) is multiplied by 180/π to convert from radians to degrees.

Arc of a Circle is a part of the circumference of a circle or we can also say the Arc of a Circle is some percentage of the circle’s circumference. As we know, a circle is defined as a two-dimensional geometrical object where all the points are equidistant from the center and the distance measured around the circle is known as a circumference and some portion of the circumference taken at a time is known as the Arc of a Circle.

In this article, we will learn the Arc of a Circle in detail, including its definition, types, and arc length formula. Other than that we will also discuss the angle subtended by an arc and the theorem related to this angle as well.