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Searching for ## Trigonometric Ratios and Circles

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Explore the full path to learning Trigonometric Ratios and Circles

Geometry

Learners calculate the length of an arc in a circle, the radius of the circle, or the measure of a central angle.

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After completing this tutorial, you will be able to complete the following:

- Calculate the arc length in a circle when the central angle is given in degrees.
- Calculate the arc length in a circle when the central angle is given in radians.
- Calculate the arc length, radius or central angle of a circle when two of them are given.

The circumference of the circle

The circumference of a circle is the complete distance around a circle.

Depending on which information you have available, you can calculate the circumference of a circle with either one:

, where C is the circumference, d is the diameter, and r is the radius of the circle.

The arc length is a part or portion of the circumference

The arc length is defined as the distance along the curved line making up an arc; it is a portion of the circumference. The measure of an arc is equal to the measure of its central angle. Therefore, if the central angle subtending an arc is then the measure of the arc is .

Calculating the arc length of the circle when the central angle is in degrees:

To find the arc length of a circle of radius 17 centimeters and central angle measure of :

Step 1: Find the circumference of the circle.

Using the formula for circumference of a circle C = 2?r , and approximating ? = 3.14, the circumference of the circle is approximately 106.76 cm.

Step 2: Use the proportional relationship.

Or since the arc length of a circle is just a portion of the circle's circumference, multiply the circumference by the ratio of the central angle of the sector to 360 (degrees in a circle):

Calculating the arc length of the circle when the central angle is in radians:

To find the arc length of a circle of radius 2 centimeters and central angle measure of radians.

Step 1: Find the area of the circle

Using the formula for circumference of a circle C = 2?r , and approximating ? = 3.14, the circumference of the circle is approximately 12.56 cm.

Step 2: Use the proportional relationship.

Or since the arc length of a circle is just a portion of the circle's circumference, multiply the circumference by the ratio of the central angle of the sector to 2?:

For the same problem, arc length of the circle can be calculated as:

NOTE: These formulas can also be used to find the radius of the circle, or the central angle when given the other information.

Approximate Time | 45 Minutes |

Pre-requisite Concepts | calculate the circumference of a circle, the concept of the central angle, concept of proportion |

Course | Geometry |

Type of Tutorial | Procedural Development |

Key Vocabulary | arc, arc length, central angle |