In Mathematics, an “arc” is a smooth curve joining two endpoints. In general, an arc is one of the portions of a circle.  It is basically a part of the circumference of a circle.

Arc is a part of a curve. An arc can be a portion of some other curved shapes like an ellipse but mostly refers to a circle. In this article, let us discuss the arc of a circle, measures and arc length formula in a detailed way.

Also, read:

• Circle and It’s Parts

What is the Meaning of Arc?

In Mathematics, an arc means, a part of a curve or the portion of a circle. The straight line joining the ends of the arc is called the chord of the circle.

Arc of a Circle

The arc of a circle is defined as the part or segment of the circumference of a circle.  A straight line that could be drawn by connecting the two ends of the arc is known as a chord of a circle. If the length of an arc is exactly half of the circle, it is known as a semicircular arc.

Symbol of Arc

In Euclidean geometry, the arc is symbolized by ‘⌒’ or ‘⌢’. The arc in the above figure is called arc AB or BA since the order of points doesn’t matter. This can be expressed as the letter AB with a curved line above it, such as \(\widehat{AB}\) and read as “arc AB”.

Measures of an Arc

The arc can be measured using two different way. They are:

• Angle of the arc
• The length of the arc

The arc’s length is computed in distance units, such as centimetres. To indicate it, the arc is preceded by the lowercase letter L(for ‘length’). For instance, \(\widehat{lAB}\) = 7 inches is read as “the length of the arc AB is 7 inches”.

Angle of the arc

The angle subtended by the arc at the center of the circle is the angle of the arc. It is denoted by \(\widehat{mAB}\), A and B are the endpoints of the arc. 

With the help of the arc length formula, we can find the measure of arc angle.

Arc length = C(θ/360°)

θ = (Arc length/C)360°

Arc Length Formula

The angle that is created by the arc at the middle of the circle is nothing but the angle measure. It’s described by the letter m preceding the name. For instance, \(\widehat{mAB}\) = 60⁰is read as “the arc AB has a measure of 60 degrees”.

Therefore, the arc length formula is given by:

When the central angle is measured in degrees, the arc length formula is:

Arc length = 2πr(θ/360)

where,

θ indicates the central angle of the arc in degrees

r indicates the radius of the arc

Since, we know, 

Circumference of the circle = 2πr

Therefore, length of the arc = C(θ/360°)

When the angle is in radians

When the central angle is in radians, the arc length formula is:

Arc length = r. θ

Where,

θ indicates the central angle of the arc in radians.

r indicates the radius of the arc.

Solved Examples

Question 1: If the angle formed by an arc is π/4 in a circle with radius equal to 3 unit. What is the length of the arc?

Answer: As we know,

Arc length = 2πr(θ/360)

Given, θ = π/4 and radius = 3 cm

Arc length = 2πr(π/4)/360)

= 2πr(π/4)/2π

= πr/4

= ¾ π unit.

Question 2: The radius of the circle is 15 cm and the arc subtends 75° at the center. What is the length of the arc?

Answer: By the formula of circumference we know that,

Circumference of circle = 2πr

C=2π x 15cm=30πcm

Length of arc = (θ/360) x C = (75°/360°)30π = 75π/12 = 25π/4 cm

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