A circular sector has central angle 70 degrees and radius 12 yd. Find the arc length s and the sector area A, rounded to three decimal places.

Step-by-step solution

1. 1

   Convert the angle to radians first. The formulas $s=r\theta$ and $A=\tfrac12 r^2\theta$ are only valid with $\theta$ in radians, because they come from taking the fraction $\theta/(2\pi)$ of a full circle. Multiply by $\pi/180$:

   $\theta=70^\circ\times\frac{\pi}{180}=\frac{7\pi}{18}\ \text{rad}$

   Using the degree measure directly would inflate both answers by a factor of about $57$.

2. 2

   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 $\pi r^2$.

3. 3

   Compute the arc length.

   $s=r\theta=12\left(\frac{7\pi}{18}\right)=\frac{14\pi}{3}\approx 14.661\ \text{yd}$

4. 4

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

   $A=\frac12 r^2\theta=\frac12(12)^2\left(\frac{7\pi}{18}\right)=72\left(\frac{7\pi}{18}\right)=28\pi\approx 87.965\ \text{yd}^2$

5. 5

   State both rounded answers.

   $\boxed{s\approx 14.661\ \text{yd},\qquad A\approx 87.965\ \text{yd}^2}$

6. 6

   Check with the relation A = ½rs. Eliminating $\theta$ between the two formulas gives $A=\tfrac12 rs$. In exact form $\tfrac12(12)\left(\tfrac{14\pi}{3}\right)=28\pi$, which is exactly the area found above. (Using the rounded $s=14.661$ instead gives $87.966$, differing in the last digit purely because of the rounding.)