Find the area of a sector of a circle of radius 3 metres with central angle 120 degrees, rounded to three decimal places.

Step-by-step solution

1. 1

   Convert the angle to radians before using the formula. $A=\tfrac12 r^{2}\theta$ is derived from the fraction $\theta/(2\pi)$ of a full circle, so it is only valid in radians:

   $\theta=120^{\circ}\times\frac{\pi}{180}=\frac{2\pi}{3}\ \text{rad}$

   Substituting $120$ directly would overstate the area by a factor of about $57$.

2. 2

   Square the radius.

   $r^{2}=3^{2}=9$

3. 3

   Apply the sector-area formula.

   $A=\frac12(9)\left(\frac{2\pi}{3}\right)=\frac{9\cdot 2\pi}{6}=3\pi$

4. 4

   Convert to a decimal.

   $A=3\pi=9.42478\approx 9.425\ \text{m}^{2}$

   $\boxed{A=3\pi\approx 9.425\ \text{m}^{2}}$

5. 5

   Check with the fraction-of-a-circle argument. $120^{\circ}$ is exactly one third of $360^{\circ}$, and the full circle has area $\pi r^{2}=9\pi$. One third of $9\pi$ is $3\pi$ ✓ — a useful shortcut whenever the angle divides $360^{\circ}$ evenly.