Circumference, Arc Length, and Radians

Circumference

• The circumference of a circle is the distance around it.

• It can be calculated using the formula $C = 2πr$, where $r$ is the radius.

• Alternatively, it can be calculated using the formula $C = πd$, where $d$ is the diameter.

Example 1

Given: Radius $r = 5 cm$
Find: Circumference $C$

$C = 2πr$
$C = 2π(5)$
$C = 10π$
$C ≈ 31.4 cm$

Circumference using Diameter

• The circumference can be found using the diameter.

• If the diameter $d = 11 cm$, then
  $C = dπ$
  $C = 11π$
  $C ≈ 34.56 cm$

Arc Length

• Arc length is a part of the circumference (length, not the arc measure).

• It is a "stretched out" or curved part of the circle.

• Arc Length $AB = \frac{m\widehat{AB}}{360°} * 2πr$

  • $m\widehat{AB}$ is the measure of arc $AB$ in degrees.

  • $r$ is the radius of the circle.

  • This formula calculates the length of the arc rather than the arc's angular measure.

Example 2

Given: $m\widehat{AB} = 75°$, Radius $r = 20 m$
Find: Arc Length of $AB$

Arc Length $= \frac{75}{360} * 2π(20)$
Arc Length $= \frac{75}{360} * 40π$
Arc Length $≈ 26.18 m$

Relating Arc Length to Circumference

• Arc Length of $AB = \frac{m\widehat{AB}}{360} * C$

• Given an arc length of 6.82 cm, and $m\widehat{AB} = 75°$, find the circumference.
  $\frac{75}{360} = \frac{6.82}{C}$
  $75C = 6.82 * 360$
  $75C = 2455.2$
  $C = \frac{2455.2}{75}$
  $C ≈ 32.74 cm$

Converting Degrees to Radians

• Radians can be found by multiplying degrees by $\frac{π}{180}$.

• $Radian = (degrees) * (\frac{π}{180})$

Example 3

Convert 30° to radians:

$Radian = 30 * (\frac{π}{180})$
$Radian = \frac{30π}{180}$
$Radian = \frac{π}{6}$