Arc

     Arc

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.

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.

$\begin{array}{l}\text{This can be expressed as the letter AB with a curved line above it, such as }\widehat{��}\\ \text{and read as “arc AB”.}\end{array}$

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’).

$\begin{array}{l}\text{For instance, }\widehat{���}\text{ = 7 inches is read as “the length of the arc AB is 7 inches”.}\end{array}$

Angle of the arc

The angle subtended by the arc at the center of the circle is the angle of the arc.

$\begin{array}{l}\text{It is denoted by }\widehat{���},\text{ A and B are the endpoints of the arc.}\end{array}$

 

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.

$\begin{array}{l}\text{For instance, }\widehat{���}={60}^0\text{ is read as “the arc AB has a measure of 60 degrees”.}\end{array}$

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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