Question

in the figure, angle zyx is measured in degrees. the area of the shaded sector can be determined using the formula \\(\frac{m\angle zyx}{360^{\circ}}(\pi r^{2})\\).

which best explains the formula?

• the central angle measure of the sector divided by the total angle measure of a circle multiplied by the area of the circle will yield the area of the sector
• the central angle measure of the sector divided by the total angle measure of a circle multiplied by the circumference of the circle will yield the area of the sector.
• the central angle measure of the sector multiplied by the area of the circle will yield the area of the sector
• the central angle measure of the sector multiplied by the circumference of the circle will yield the area of the sector

Explanation:

To determine the correct explanation, recall the formula for the area of a sector. The area of a circle is \( \pi r^2 \), and a sector is a fraction of the circle based on its central angle. The total angle in a circle is \( 360^\circ \). The formula \( \frac{m\angle ZYX}{360^\circ}(\pi r^2) \) takes the ratio of the sector’s central angle (\( m\angle ZYX \)) to the total circle angle (\( 360^\circ \)) and multiplies it by the circle’s area (\( \pi r^2 \)).

• The second option is incorrect because it mentions circumference (used for arc length, not sector area).
• The third option is incorrect as it multiplies the central angle by the circle’s area (not a ratio).
• The fourth option is incorrect as it involves circumference and multiplication (not the correct sector area logic).

Only the first option correctly describes the formula: the sector’s central angle divided by the circle’s total angle, multiplied by the circle’s area, gives the sector’s area.

Answer:

The central angle measure of the sector divided by the total angle measure of a circle multiplied by the area of the circle will yield the area of the sector