length of an arc of a sector of a circle with radius r and angle with degree measure theta is
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.
- The circumference of a circle is given by 2\pi r. This represents the total distance around the circle.
- In a full circle (360 degrees), we have the entire circumference.
- 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.
- Since the angle \theta is given in degrees, we divide it by 360 to find the fraction of the circle it represents.
- 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.