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.

Part 1 is the number of a certain color (blue) of M&Ms and Whole 1 is the total number of 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 = 2r .
• 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 ______ 25x .
• The products are equal. Solve for x. ______251.5 = 25x ⇒ 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 sector360° 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 = 2r = 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 = 8x
• 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