NDA 2019 Maths · Mathematics · Limits · medium

A regular polygon with n sides is inscribed in a circle of radius 1. As n→∞, the perimeter of the polygon tends to

Single correct option

A regular polygon with

n

sides is inscribed in a circle of radius

1

. As

n\to\infty

, the perimeter of the polygon tends to

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Answer: (D) $2\pi$

Each side subtends an angle

\frac{2\pi}{n}

at the centre, so its length is

2\sin\frac{\pi}{n}

and the perimeter is

2n\sin\frac\pi n=2\pi\cdot\frac{\sin(\pi/n)}{\pi/n}

. As

n\to\infty

the ratio tends to

1

, giving

2\pi

, the circumference. The value

\pi

is the semicircle, and

\infty

is a false conclusion from

n\to\infty

Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake