Question

If a regular hexagon is inscribed in a circle of radius r, what is its perimeter?
A. 3r
B. 6r
C. 9r
D. 12r

Answer 1

- Hint: Given the regular hexagon is inscribed in a circle of radius r. Try to prove that a hexagonal is made of six equilateral triangles by joining the opposite sides of the hexagon.
After that you will observe that the sides of each equilateral triangle is r, equal to the radius of the circle. Now, you can easily find the perimeter of the hexagon.

Complete step-by-step solution -

The regular hexagon is inscribed in a circle of radius r.
So, it is inside the circle.
By joining opposite sides of the hexagon, it forms six (6) central angles at centre O each of which $=\dfrac{360{}^\circ }{6}=60{}^\circ $.
And, you see the six triangles are formed.
The two sides of each triangle are the radius of the circle and thus are equal.
\[\therefore \] The base angles of every triangle are equal.
Since, the central angle is $60{}^\circ $.
This gives,
$\Rightarrow Base\ angles\ =\dfrac{120{}^\circ }{6}=60{}^\circ $
\[\therefore \] The triangles are equilateral triangles.
This gives, all sides are equal.
You can observe this in the figure drawn below.
\[\therefore \] All sides of each triangle are r.
Hence,
Perimeter of regular hexagon $=6\times side\ of\ equilateral\ triangle$
$\begin{align}
  & =6\times r \\
 & =6r \\
\end{align}$
Hence, option B is correct.

Note: This question can be solved easily in this way - (You can do this question by observing and thinking in this way).
A regular hexagon is a polygon whose all sides are equal and it has been made of six equilateral triangles.
Now, if a regular hexagon is inscribed in a circle then its side is equal to the radius of the circle.
Hence, the perimeter of regular hexagon = 6r.