Question

The angle of the sector of a circle whose area is one-sixteenth of the area of the circle is _______.

Answer 1

22.5°

Answer 2

The area of a sector of a circle is proportional to its central angle. The formula relating the area of a sector to the area of the circle is:

$\frac{\text{Area of Sector}}{\text{Area of Circle}} = \frac{\theta}{360^\circ}$

where $\theta$ is the central angle of the sector in degrees.

Given that the area of the sector is one-sixteenth of the area of the circle, we can write:

$\text{Area of Sector} = \frac{1}{16} \times \text{Area of Circle}$

Substitute this into the formula:

$\frac{\frac{1}{16} \times \text{Area of Circle}}{\text{Area of Circle}} = \frac{\theta}{360^\circ}$

The "Area of Circle" term cancels out from both sides:

$\frac{1}{16} = \frac{\theta}{360^\circ}$

Now, solve for $\theta$:

$\theta = \frac{1}{16} \times 360^\circ = 22.5^\circ$

The angle of the sector is $22.5^\circ$.