Question

which statements are true regarding the area of circle d? select two options. the area of the circle depends on the square of the radius. the area of circle d is 36π cm². the area of circle d is 324π cm². the area of the circle depends on the square of pi. the area of the circle depends on the central angle.

Explanation:

Step1: Recall 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 of the circle. From this formula, we can see that the area depends on the square of the radius. So the statement "The area of the circle depends on the square of the radius" is true.

Step2: Determine the radius of circle D

From the diagram, the length from the center \( D \) to the point \( C \) (the radius) is \( 18\space\text{cm} \).

Step3: Calculate the area of circle D

Using the formula \( A = \pi r^{2} \), substitute \( r = 18\space\text{cm} \) into the formula. We get \( A=\pi\times(18)^{2}=\pi\times324 = 324\pi\space\text{cm}^2 \). So the statement "The area of circle D is \( 324\pi\space\text{cm}^2 \)" is true.

Now let's analyze the other statements:

• The statement "The area of circle D is \( 36\pi\space\text{cm}^2 \)" is false because when we calculated the area with \( r = 18 \), we got \( 324\pi \), not \( 36\pi \).
• The statement "The area of the circle depends on the square of pi" is false because the formula is \( A=\pi r^{2} \), and \( \pi \) is a constant, the area depends on the square of the radius, not the square of \( \pi \).
• The statement "The area of the circle depends on the central angle" is false because the area of a circle (not a sector) is a property of the radius (or diameter) of the circle and does not depend on the central angle (central angle is related to the area of a sector, not the whole circle).

Answer:

The area of the circle depends on the square of the radius.
The area of circle D is \( 324\pi\space\text{cm}^2 \)