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Arc Of A Circle




 
In these lessons, we will learn

  • the arc of a circle
  • the central angle
  • the arc measure
  • the arc length formula when the arc measure is given in degrees
  • the arc length formula when the arc measure is given in radians
  • how to calculate the arc length using the arc length formula
Related Topics: More Geometry Lessons

Arc of a Circle

An arc is any connected part of the circumference of a circle.

arc

In the diagram above, the part of the circle from B to C forms an arc. It is called arc BC.

An arc could be a minor arc, a semicircle or a major arc.

  • A semicircle is an arc that is half a circle.
  • A minor arc is an arc that is smaller than a semicircle.
  • A major arc is an arc that is larger than a semicircle.

Central Angle

A central angle is an angle whose vertex is at the center of a circle.

arc

In the diagram above, the central angle for arc BC is 45°.

The sum of the central angles in any circle is 360°.



Arc Measure

The measure of a semicircle is 180°.

The measure of a minor arc is equal to the measure of the central angle that intercepts the arc. We can also say that the measure of a minor arc is equal to the measure of the central angle that is subtended by the arc. In the diagram below, the measure of arc BC is 45°,

arc

The measure of the major arc is equal to 360° minus the measure of the associated minor arc.

The following video shows how to identify semicircle, minor arc and major arc and their measures.



 

Arc Length Formula

The arc length is the distance along the part of the circumference that makes up the arc.


Arc Measure given in Degrees

Since the arc length is a fraction of the circumference of the circle, we can calculated it in the following way. Find the circumference of the circle and then multiply by the measure of the arc divided by 360°. Remember that the measure of the arc is equal to the measure of the central angle.

The formula for the arc length of a circle is

arc length

where r is the radius of the circle and m is the measure of the arc (or central angle) in degrees.

Worksheet to calculate arc length and area of a sector (degrees).

Arc Measure given in Radians

If the measure of the arc (or central angle) is given in radians, then the formula for the arc length of a circle is the product of the radius and the arc measure.

Arc Length = r × m

where r is the radius of the circle and m is the measure of the arc (or central angle) in radians

The above formulas allow us to calculate any one of the values given the other two values.

Worksheet to calculate arc length and area of sector (radians).

Calculate Arc Length given Measure of Arc in degrees

From the formula, we can calculate the length of the arc.

Example:

calculate arc length

If the circumference of the following circle is 54 cm, what is the length of the arc ABC?

Solution:

Circumference = 2πr = 54

arc length

Example:

If the radius of a circle is 5 cm and the measure of the arc is 110˚, what is the length of the arc?

Solution:

arc length


 

Videos - Arc Length - Degrees

These video shows how to define arc length and how it is different from arc measure. They will also show how to calculate the length of an arc when the arc measure is given in degrees.

Arc Length Theorem - How to Find the Length of an Arc
This video shows how to use the Arc Length Formula when the arc measure is given in degrees.
This video lesson discusses how to find the length of an arc. First, the arc length theorem is reviewed and explained. An example of find the length of a major arc is modeled. The given information is the measure of the related minor arc and the radius of the circle.
The following video shows how to find the arc length on a circle when the central angle is given in degrees.


Calculate Arc Length given Measure of Arc in radians

If the measure of the arc (or central angle) is given in radians, then the formula for the arc length of a circle is

Arc Length = r × m

where r is the radius of the circle and m is the measure of the arc (or central angle) in radians

This video shows how to use the Arc Length Formula when the measure of the arc is given in radians.
Definition of Arc Length and Finding the Arc Length when the central angle is given in radians.


 

Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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