Question

Which statements are true regarding the area of circles and sectors? Check all that apply.

• A. The area of a circle depends on the length of the radius.
• B. The area of a sector depends on the ratio of the central angle to the entire circle.
• C. The area of a sector depends on $\mathrm{\pi }$.
• D. The area of the entire circle can be used to find the area of a sector.
• E. The area of a sector can be used to find the area of a circle.

Answer 1

Answer: (A), (B)

The area of a sector depends on the ratio of the central angle to the entire circle.

Explanation for the correct option:

Option (A):

It states that “The area of a circle depends on the length of the radius”. Recall that the area of a circle is given by $\mathrm{\pi }{r}^2$, where $r$ is its radius. So, the area of a circle depends on its radius and this statement is true. Thus, option A is correct.

Option (B):

It states that “The area of a sector depends on the ratio of the central angle to the entire circle”. Recall that the area of a sector is given by $\frac{\theta }{360°}\times \mathrm{\pi }{r}^2$, where $r$ is its radius, and $\theta$ is its central angle in degrees. Since $\frac{\theta }{360°}$ represents the ratio of the central angle of the sector to the central angle of the entire circle, the area of the sector depends on it. Thus, option B is correct.

Explanation for the in-correct option:

Option (C):

It states that "The area of a sector depends on $\mathrm{\pi }$". $\mathrm{\pi }$ is the ratio of circumference to diameter for any circle. It is an unchanging constant of nature. It is not some variable whose value can be changed to change the area of a sector. Since its value is a known constant, other values cannot depend on it. Thus, option C is incorrect.

Option (D):

It states that “The area of the entire circle can be used to find the area of a sector”. Recall that the area of a sector is given by $\frac{\theta }{360°}\times \mathrm{\pi }{r}^2$, where $r$ is its radius, and $\theta$ is its central angle in degrees. Also, the area of a circle is given by $\mathrm{\pi }{r}^2$, where $r$ is its radius. So, it is not possible to find the area of a sector $\frac{\theta }{360°}\times \mathrm{\pi }{r}^2$, by knowing only the area of the circle $\mathrm{\pi }{r}^2$, the central angle of the sector $\theta$ is also required. Thus, option D is incorrect.

Option (E):

It states that “The area of a sector can be used to find the area of a circle”. Recall that the area of a sector is given by $\frac{\theta }{360°}\times \mathrm{\pi }{r}^2$, where $r$ is its radius, and $\theta$ is its central angle in degrees. Also, the area of a circle is given by $\mathrm{\pi }{r}^2$, where $r$ is its radius. Thus, the area of a sector is $\frac{\theta }{360°}$ times the area of a circle. This means that the area of a circle is $\frac{360°}{\theta }$ times the area of the sector. So, it is not possible to find the area of a circle $\mathrm{\pi }{r}^2$, by knowing only the area of the sector $\frac{\theta }{360°}\times \mathrm{\pi }{r}^2$, the central angle of the sector $\theta$ is also required. Thus, option E is incorrect.

Hence, options A, and B are correct.