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. For "The area of a circle depends on the length of the radius": The formula for the area of a circle is \( A = \pi r^2 \), where \( r \) is the radius. So the area is dependent on the radius, this statement is true.
2. For "The area of a sector depends on the ratio of the central angle to the entire circle": The formula for the area of a sector is \( A=\frac{\theta}{360^\circ}\times\pi r^2 \) (for a circle with \( 360^\circ \) and central angle \( \theta \)), so it depends on the ratio \( \frac{\theta}{360^\circ} \), this statement is true.
3. For "The area of a sector depends on pi": Since the formula for the area of a sector includes \( \pi \) (from the circle's area formula which the sector is a part of), this statement is true.
4. For "The area of the entire circle can be used to find the area of a sector": As the sector area is a fraction (based on central angle ratio) of the circle's area, we can use the circle's area (\( \pi r^2 \)) and multiply by the ratio of the central angle to \( 360^\circ \) to get the sector area, this statement is true.
5. For "The area of a sector can be used to find the area of a circle": A single sector's area is a part of the circle, but without knowing the ratio of the central angle (or having multiple sectors with known angles to sum up to \( 360^\circ \)), we can't generally find the circle's area from one sector's area. So this statement 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.