Question

Prove that in a regular hexagon, the sum of all its interior angles is twice the sum of exterior angles formed by producing the sides in the same order.Find the value of each interior and the corresponding exterior angle of the regular hexagon.

Answer 1

The value of each interior angle is 120°

The value of each exterior angle is 60°

The proof is given below:

We know that sum of all the internal angles of n sided polygon is given by

$S=(n-2)\times 180^\circ$

If the polygon is a regular polygon, then each angle is given by

$A=\frac{S}{n}$

For a regular hexagon, sum of all the interior angles

$S=(6-2)\times 180^\circ$

$\implies S=720^\circ$   ........ (1)

Each interior angle is $\frac{720^\circ}{6}=120^\circ$

If the interior angles is known then each exterior angle is given by

$180^\circ-\text{ Interior Angle}$

Thus, each exterior angle of regular hexagon is

$=180^\circ-120^\circ$

$=60^\circ$

Thus, sum of all the exterior angles

$S'=6\times 60^\circ$

$\implies S'=360^\circ$   ......... (2)

From (1) and (2) we can see

$S=2S'$                                             (Proved)

Hope this answer is helpful.

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