Question

which statements are true regarding the area of circles and sectors? check all that apply.
the area of a circle depends on the length of the radius.
the area of a sector depends on the ratio of the central angle to the entire circle.
the area of a sector depends on pi.
the area of the entire circle can be used to find the area of a sector.
the area of a sector can be used to find the area of a circle.

Explanation:

1. The formula for the area of a circle is \( A = \pi r^2 \), so the area depends on the radius (True).
2. The area of a sector is \( A = \frac{\theta}{360^\circ} \times \pi r^2 \) (where \(\theta\) is the central angle), so it depends on the ratio of the central angle to the full circle (True).
3. Since the sector area formula includes \(\pi\), it depends on \(\pi\) (True).
4. Using the sector area formula, if we know the area of the full circle (\(\pi r^2\)) and the central angle ratio, we can find the sector area (True).
5. To find the circle's area from a sector, we need to know the central angle (not always possible from just the sector area), so this is False.

Answer:

• The area of a circle depends on the length of the radius.
• The area of a sector depends on the ratio of the central angle to the entire circle.
• The area of a sector depends on pi.
• The area of the entire circle can be used to find the area of a sector.