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:

Step1: Analyze the formula for the area of a circle

The formula for the area of a circle is \(A = \pi r^{2}\), where \(r\) is the radius. So, the area of a circle depends on the length of the radius.

Step2: Analyze the formula for the area of a sector

The formula for the area of a sector is \(A_{sector}=\frac{\theta}{360^{\circ}}\times\pi r^{2}\) (when \(\theta\) is in degrees) or \(A_{sector}=\frac{1}{2}r^{2}\alpha\) (when \(\alpha\) is in radians). The area of a sector depends on the ratio of the central angle (\(\theta\) or \(\alpha\)) to the entire circle (\(360^{\circ}\) or \(2\pi\) radians), and it also depends on \(\pi\) (since the formula is related to the area of the circle which has \(\pi\) in it). Also, if we know the area of the entire circle \(A = \pi r^{2}\), and the ratio of the sector's central angle to the full - circle angle, we can find the area of the sector. Conversely, if we know the area of a sector and the ratio of its central angle to the full - circle angle, we can find the area of the circle.

Answer:

All statements are true.