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The side of a regular hexagon is 2 cm. Then, what is the ratio of the radius of the circumscribed circle to the radius of the inscribed circle?

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23

3

2

13

answer is A.

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Detailed Solution

Let the diagram of the given situation be,

From the above figure, EO is the diameter of the circumscribed circle.
In triangle EMF, we have

sin 30 ∘ = E M E F (∵ ∠ E F M = 30 ∘) 12 = E M 2 E M = 1

O is the center of the hexagon. Then,

O M = A F 2 (∵ A F = 2) O M = 1

Then, the radius of the circumscribed circle is,

O E = E M + O M = 1 + 1 = 2

From the figure, we can say that PO is the radius of the inscribed circle. Since

P O = F M

, then

cos 30 ∘ = F M E F 32 = F M 2 F M = 3

The ratio of the radius of the circumcircle and incircle is given by

O E P O = 23

.
Hence, the correct option is 1.

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