In this section, the circumference is found when the arc length, radius, and sector degree measure is given. A calculator is needed.

The proportion used for arc length is:

Degree measure of sector 360° ⇒ Degree measure of circle

The same proportion is used when the arc length is given and the missing piece of information is the circumference of the circle. The variable will be in the denominator of the fraction instead of the numerator .

**Example**: Find the circumference of the circle below.

Answer the following questions using the variable *x* or *y* for unknown values. Click on the blanks to reveal the answers.

- The degree measure of the sector is ______. 60°
- The degree measure of the circle is ______. 360°
- The measure of the arc of the circle is ______. 7 cm
- The unknown is the ______. circumference
- Substitute the values into the proportion
Degree measure of sector
Degree measure of circle
=
Arc length
Circumference
=
______.
60
360
=
7
*x* - Find the cross product and solve for
*x*. ______. 60 360 = 7*x*⇒ 1 6 = 7*x*⇒*x*= 42 cm - The ______ circumference of the circle is ______. 42 cm

The same proportion is used when the missing piece of information is the sector measure of the circle.

**Example**: Find the sector measure of the circle below.

Answer the following questions using the variable *x* or *y* for unknown values. Click on the blanks to reveal the answers.

- The unknown or the variable
*x*, is the ______. sector measure - The degree measure of the circle is ______. 360°
- The measure of the arc of the circle is ______. 3 cm
- Substitute the values into the proportion
Degree measure of sector
Degree measure of circle
=
Arc length
Circumference
=
______.
*x*360 = 3 10 - Find the cross product and solve for
*x*. ______.*x*360 = 3 10 ⇒ 10*x*= 1080 ⇒*x*= 108° - The ______ arc or sector measure of the circle is ______. 108°

The proportion Degree measure of sector 360° ⇒ Degree measure of circle = Arc length Circumference is used to find the arc length, circumference, and degree measure of the sector.