Question

A regular hexagon is inscribed in a circle of radius \[r\]. The perimeter of regular hexagon is
A. \[3r\]
B. \[6r\]
C. \[9r\]
D. \[12r\]

Answer 1

Hint: In this question, first draw the diagram it will give us a clear picture of what we have to find out. The perimeter of a Regular hexagon is just the sum of all 6 sides. So, use this concept to reach the solution of the given problem.

Complete step-by-step answer:
The regular hexagon is inscribed in a circle of radius \[r\].
So, it is inside the circle. The diagram will be like this:

By joining opposite sides of a hexagon, it forms 6 central angles at centre each of which equals to \[ = \dfrac{{{{360}^\circ}}}{6} = {60^\circ}\].
And the six triangles are formed.
The two sides of each triangle are the radius of the circle and both are equal.
Therefore, the base angles of every triangle are equal. \[\left[ {\because {\text{central angle is 6}}{{\text{0}}^\circ}} \right]\]
So, base angle \[ = \dfrac{{{{120}^\circ}}}{2} = {60^\circ}\]
Therefore, the triangles are equilateral triangles.
So, here all sides are equal for an equilateral triangle.
Therefore, all sides of each triangle are equal to \[r\].
So, perimeter of regular hexagon \[ = 6 \times side = 6r\]
Hence, correct option is B. \[6r\]

Note: A regular hexagon has six sides and six angles. Lengths of all the sides and the measurement of all the angles are equal. The total number of diagonals in a regular hexagon is 9. The sum of all interior angles is equal to \[{720^\circ}\] which each interior angle measures \[{120^\circ}\].