In this section, a proportion is used to find the arc length. A calculator is needed.

A proportion written in the form Part 1 Whole 1 = Part 2 Whole 2 can be used to find the length of an arc. Part 1 and Whole 1 are usually what we know. Part 2 and Whole 2 include one unknown piece of information.

Proportion and Cross Product Review

**Example**: Use the information obtained in section one to find the arc length of the blue M&Ms on the circle.

Below is a chart with the information obtained in section 1.

Color |
Number |
Fraction of Total |

Red | 6 | 6 25 |

Blue | 5 | 5 25 = 1 5 |

Green | 6 | 6 25 |

Yellow | 4 | 4 25 |

Brown | 4 | 4 25 |

Total | 25 | 25 25 = 1 |

Part 1 is the number of a certain color (blue) of M&Ms and Whole 1 is the total number of M&Ms.

Part 1 Whole 1 = Number of a certain color of M&M All the M&Ms

The variable, or information that we don't know, is part of the second fraction: Part 2 Whole 2

Three out of the four pieces of information must be known in order to solve a proportion.

Fill in the following. Use the variable *x* or *y* for unknown values. Click on the blanks to reveal the answers.

- Part 1 is the number of blue M&Ms ______. 5
- Whole 1 is the total number of M&Ms ______. 25
- Part 2 is the arc length of the blue M&Ms
______.
*x* - The circumference of the circle is ______ Whole 2 .
- The formula for the circumference of a circle is
______
*d*= 2*r*. - The circumference needs the value of the ______ diameter or ______ radius .
- The radius of the M&M circle is 8 cm. The circumference is ______ 2(8) = 16 = 50.3 cm .
- Multiply the denominator of the first
ratio
by the numerator of the second ratio
______
25
*x*. - The products are equal. Solve for
*x*. ______251.5 = 25*x*⇒*x*= 251.5 25 = 10.1cm . - The arc length has a measure of ______. 10.1cm

A general equation to find the length of an arc is:

*Degree measure of the sector*360° *in a circle*
=
*Measure of arc length*
*Circumference of circle*

**Example:** Given the circle below, find the arc length of the shaded part of arc *AB*.

- The central angle measure of the sector is ______. 45°
- There are ______. 360°
- The radius of the circle is
______.
*r*= 8*units* - The arc length is the variable
______.
*x*or*y* - The circumference of the circle is
______.
*C*= 2*r*= 2(8) = 16 = 50.2cm - The proportion to find the arc length is
______,
45
360
=
*x*50.2 simplify. ______. 1 8 =*x*50.2 - Find the cross product.
______.
50.2 = 8
*x* - Solve for the arc length. ______. 6.3 cm

Check logically:

- The fraction of the circle that is the sector is the ratio ______. 45 360 = 1 8
- The circumference of the circle is about ______. 50.2 cm
- 1 8 of the circumference is about ______ 6 cm since ______ 6 * 8 = 48 and our circumference is 50.
- Our answer was 6.3 cm, a little more than ______. 6 cm

Another Example