Question

If the radius of a circle is $r,$ then:

• A. the value of the perimeter of a sector is always greater than $2r$
• B. the value of the perimeter of a sector is always greater than $2\pi r$
• C. the value of the perimeter of a sector is always less than $2\pi r$
• D. the value of the perimeter of a sector is always less than $2r+2\pi r$

Answer 1

Answer: (A), (D)

the value of the perimeter of a sector is always less than $2r+2\pi r$

The formula for the perimeter of a sector is $2r+\frac{\theta }{{360}^{\circ }}\times 2\pi r,$ where $\theta$ is the angle subtended by the sector at the centre of the circle

The least possible value of $\theta$ is $0^{\circ }.$ But at $\theta =0^{\circ },$ there will be no sector. So $\theta$ can attain any value greater than $0.$ At $\theta =0$ the value of the perimeter of the sector is $2r.$ So, the value of the perimeter of a sector is always greater than $2r.$

The maximum possible value of $\theta$ is ${360}^{\circ }.$ But at $\theta ={360}^{\circ },$ it will be a complete circle. So $\theta$ can attain any value less than $360.$ At $\theta =360$ the value of the perimeter of the sector is $2r+\frac{{360}^{\circ }}{{360}^{\circ }}\times 2\pi r$ i.e. $2r+2\pi r$ So, the value of the perimeter of a sector is always less than $2r+2\pi r.$

Options (a.) and (d.) are correct choices.