A circular sector has central angle pi/3 radians and radius 2 ft. Find the arc length s and the sector area A, rounded to three decimal places.

Step-by-step solution

1. 1

   Write the two sector formulas. For a central angle $\theta$ in radians,

   $s=r\theta,\qquad A=\frac12 r^2\theta$

   They are simply the fraction $\theta/(2\pi)$ of the full circumference $2\pi r$ and the full area $\pi r^2$.

2. 2

   Compute the arc length.

   $s=r\theta=2\left(\frac{\pi}{3}\right)=\frac{2\pi}{3}\approx 2.094\ \text{ft}$

3. 3

   Compute the sector area. Square the radius first, then halve:

   $A=\frac12 r^2\theta=\frac12(2)^2\left(\frac{\pi}{3}\right)=\frac12(4)\left(\frac{\pi}{3}\right)=\frac{2\pi}{3}\approx 2.094\ \text{ft}^2$

4. 4

   State both rounded answers.

   $\boxed{s\approx 2.094\ \text{ft},\qquad A\approx 2.094\ \text{ft}^2}$

5. 5

   Check with the relation A = ½rs. Eliminating $\theta$ gives $A=\tfrac12 rs$, and $\tfrac12(2)\left(\tfrac{2\pi}{3}\right)=\tfrac{2\pi}{3}$, exactly the area found above.

6. 6

   Why the two answers are numerically equal here. $A=\tfrac12 r^2\theta$ and $s=r\theta$ differ by the factor $\tfrac12 r$, which equals $1$ exactly when $r=2$. So this coincidence is special to a radius of $2$ and does not carry over to the other radii — and the units still differ ($\text{ft}$ versus $\text{ft}^2$).