Question

suppose the radius r is 2 units and the angle \\(\theta\\) is \\(120^\circ\\). what is the sector area a? please round to two places after the decimal point.

Explanation:

Step1: Recall the sector area formula

The formula for the area of a sector with radius \( r \) and central angle \( \theta \) (in degrees) is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \).

Step2: Substitute the given values

We know that \( r = 2 \) units and \( \theta=120^{\circ} \). Substituting these values into the formula:
\( A=\frac{120^{\circ}}{360^{\circ}}\times\pi\times(2)^{2} \)

Step3: Simplify the expression

First, simplify \( \frac{120^{\circ}}{360^{\circ}}=\frac{1}{3} \), and \( (2)^{2} = 4 \). So the expression becomes \( A=\frac{1}{3}\times\pi\times4=\frac{4\pi}{3} \).

Step4: Calculate the numerical value

Using \( \pi\approx3.14159 \), we have \( A=\frac{4\times3.14159}{3}\approx\frac{12.56636}{3}\approx4.19 \) (rounded to two decimal places).

Answer:

\( 4.19 \)