Length of an arc of a sector of a circle with radius r and angle with degree measure theta is

General
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length of an arc of a sector of a circle with radius r and angle with degree measure theta is

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The length of an arc of a sector of a circle with radius r and angle with degree measure \theta can be calculated using the formula:

\text{Length of arc} = \frac{\theta}{360} \times 2\pi r

Let me explain this formula step by step.

  1. The circumference of a circle is given by 2\pi r. This represents the total distance around the circle.
  2. In a full circle (360 degrees), we have the entire circumference.
  3. To find the length of an arc in a sector of a circle, we need to find the fraction of the circumference that corresponds to the angle \theta.
  4. Since the angle \theta is given in degrees, we divide it by 360 to find the fraction of the circle it represents.
  5. Finally, we multiply this fraction by the circumference (2$\pi r$) to find the length of the arc.

Let’s do an example:

Suppose we have a circle with a radius of 5 units and an angle with a measure of 60 degrees. To find the length of the arc in this sector, we can use the formula:

$\text{Length of arc} = \frac{60}{360} \times 2\pi \times 5$

Simplifying this expression, we get:

\text{Length of arc} = \frac{1}{6} \times 2\pi \times 5

\text{Length of arc} = \frac{\pi}{3} \times 5

\text{Length of arc} = \frac{5\pi}{3}

So, in this example, the length of the arc in the sector is \frac{5\pi}{3} units.