For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is: - Mathematics MCQ AIEEE 2026 | ExamTest.live | ExamTest.live

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For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is:

A $There is a regular polygon with \frac{r}{R} = \frac{1}{2}$
B $There is a regular polygon with \frac{r}{R} = \frac{2}{3}$
C $There is a regular polygon with \frac{r}{R} = \frac{3}{2}$
D $There is a regular polygon with \frac{r}{R} = \frac{1}{2}$

Solution & Step-by-step Explanation

For a regular n -gon, the ratio is \frac{r}{R} = cos (\frac{π}{n}) . n = 3 (Equilateral triangle): cos (π /3) = 1/2 . (A is true). n = 4 (Square): cos (π /4) = 1/ 2 . (D is true). n = 6 (Hexagon): cos (π /6) = 3 /2 . (C is true).For r / R = 2/3, we need cos (π / n) = 2/3 . Since π / n must be an angle of the form π /3, π /4, π /5 \dots, 2/3 does not correspond to a standard regular polygon n .

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For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is:

A

There is a regular polygon with \frac{r}{R} = \frac{1}{2}

B

There is a regular polygon with \frac{r}{R} = \frac{2}{3}

C

There is a regular polygon with \frac{r}{R} = \frac{3}{2}

D

There is a regular polygon with \frac{r}{R} = \frac{1}{2}

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