A sector with radius 12 cm has area 132 cm². Find the central angle.

The correct option is 105°.

To find the central angle of the sector, we use the standard formula for the area of a sector of a circle:

$\text{Area of sector}=\frac{\theta }{\mathrm{360°}}\times \pi r^2$

From the given question, we have the following values:
Radius of the sector, $r=12\text{cm}$
Area of the sector $=132{\text{cm}}^2$
Using $\pi =\frac{22}{7}$

Now, substitute these given values into the area formula:

$132=\frac{\theta }{\mathrm{360°}}\times \frac{22}{7}\times {12}^2$

Simplify the square of the radius:

$132=\frac{\theta }{\mathrm{360°}}\times \frac{22}{7}\times 144$

Rearrange the equation to solve for the central angle $\theta$:

$\theta =\frac{132\times \mathrm{360°}\times 7}{22\times 144}$

Simplify the terms step-by-step:
Divide 132 by 22:

$\frac{132}{22}=6$

Now substitute 6 back into the expression:

$\theta =\frac{6\times \mathrm{360°}\times 7}{144}$

Divide 144 by 6:

$\frac{144}{6}=24$

This simplifies our expression to:

$\theta =\frac{\mathrm{360°}\times 7}{24}$

Now divide 360° by 24:

$\frac{\mathrm{360°}}{24}=\mathrm{15°}$

Finally, multiply 15° by 7:

$\theta =\mathrm{15°}\times 7=\mathrm{105°}$

Thus, the central angle of the sector is 105°.