The Concept of Arc in Mathematics: Definition and Calculation

by Yuvi K - December 30, 2023

In mathematics, an arc is a significant concept, particularly in geometry and trigonometry. It refers to a continuous segment of a curve, typically a circle or other smooth curve.

Definition

Basic Definition: An arc on a circle is a portion of the circumference of the circle. It is the curve between two points on the circumference. In other contexts, an arc can be a segment of any curved line.

Types of Arcs:

Minor Arc: A smaller arc segment between two points on a circle.
Major Arc: A larger arc segment that represents the longer distance between two points on a circle.
Semicircle: A special case of an arc that represents exactly half of a circle.

Arc Length: This is the measure of the distance along the curved line making up the arc. It is a fraction of the circumference of the entire circle.

Calculation of Arc Length

To calculate the length of an arc in a circle, you need to know two things: the radius of the circle (r) and the central angle (θ) subtended by the arc. The central angle can be in degrees or radians, which are two different ways of measuring angles.

Formula Using Radians:

$A rc L e n g t h = r \times θ$
Here, θ is the angle in radians.

Formula Using Degrees:

$A rc L e n g t h = (θ \times 2 π r$
Here, θ is the angle in degrees.

Example Calculations

Example in Radians:

If a circle has a radius of 5 cm and the arc subtends a central angle of 2 radians, the arc length is $5 \times 2 = 10$ cm.

Example in Degrees:

For a circle with a radius of 10 cm and an arc subtending a central angle of 60 degrees,
the arc length is $60/3 60 \times 2 π \times 10 \approx 10.47$ cm.

Applications

Geometry: In studying circles and curves.
Trigonometry: In understanding the relationships between angles and sides in triangles.
Engineering and Physics: In describing trajectories, circular motions, and various other applications.
Architecture and Design: In designing curved structures and elements.

Conclusion

The concept of an arc is fundamental in mathematics, with wide applications across various fields. Understanding how to define and calculate the length of an arc is crucial for many theoretical and practical applications in science and engineering.