Question

a sector has an area of 32π square units and a radius of 8 units. find the measurement of its central angle in radian measure. use the keypad to enter your answers in the boxes to complete the solution. (square=\frac{1}{2}(square)^2cdot\theta) (\theta = square)

Explanation:

Step1: Recall sector - area formula

The formula for the area $A$ of a sector of a circle with radius $r$ and central - angle $\theta$ (in radians) is $A=\frac{1}{2}r^{2}\theta$.

Step2: Substitute given values

We are given that $A = 32\pi$ and $r = 8$. Substituting these values into the formula, we get $32\pi=\frac{1}{2}(8)^{2}\cdot\theta$.

Step3: Simplify the right - hand side

First, calculate $(8)^{2}=64$, then $\frac{1}{2}(8)^{2}=\frac{1}{2}\times64 = 32$. So the equation becomes $32\pi=32\theta$.

Step4: Solve for $\theta$

Divide both sides of the equation $32\pi=32\theta$ by 32. We have $\theta=\pi$.

Answer:

$\pi$