Discussion

If the angle of a sector is \( z^{\circ} \), then the formula used to calculate the length of its arc is

Answer: D

An arc is the fraction of the circumference that its central angle is of a full turn.

\[ \text{arc length} = \frac{z^{\circ}}{360^{\circ}} \times \text{circumference} = \frac{z}{360} \times 2\pi r \]

The same fraction applied to the area of the circle gives the sector's area instead:

\[ \text{sector area} = \frac{z}{360} \times \pi r^{2} \]

Length comes from circumference and area from area - matching the two is what the options are testing.

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