Question

The area of a regular polygon of $2n$ sides inscribed in a circle is given by?

• A. The geometric mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  
• B. The arithmetic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  
• C. The harmonic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  
• D. None of the above

Answer 1

Answer: (A)

The geometric mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

Let $a$ be the radius of the circle

Then,$s_1=$ Area of regular polygon of n sides inscribed in the circle $=\frac{1}{2}n{a}^2\sin \left(\frac{2\pi }{n}\right)$

$s_2=$ Area of regular polygon of n sides circumscribing in the circle $=n{a}^2\tan \frac{\pi }{n}$

$s_3=$ Area of regular polygon of 2n sides inscribed in the circle $=n{a}^2\tan \frac{\pi }{n}$

[replacing $n$ by $2n$ is $(S_1$]

$\therefore$ Geometric mean of $S_1$ and

$S_2$ $=\sqrt{\left(S_1{S}_2\right)}=n{a}^2\sin \left(\frac{\pi }{n}\right)=S_3$