Question

Fill in the blanks.
(i) Each interior angle of a regular octagon is (.........)°.
(ii) The sum of all interior angles of a regular hexagon is (.........)°.
(iii) Each exterior angle of a regular polygon is 60°. This polygon is a .........
(iv) Each interior angle of a regular polygon is 108°. This polygon is a .........
(v) A pentagon has ......... diagonals.

Answer 1

(i) Octagon has 8 sides.
$$\begin{gathered} \therefore \mathrm{Interior}\mathrm{angle}=\frac{180°n-360°}{n} \\ \mathrm{Int}\mathrm{erior}\mathrm{angle}=\frac{(180°\times 8)-360°}{8}=135° \end{gathered}$$
(ii) Sum of the interior angles of a regular hexagon = $(6-2)\times {180}^{\circ }={720}^{\circ }$

(iii) Each exterior angle of a regular polygon is ${60}^{\circ }$.
$\therefore \frac{360}{60}=6$
Therefore, the given polygon is a hexagon.

(iv) If the interior angle is ${108}^{\circ }$, then the exterior angle will be ${72}^{\circ }$. (interior and exterior angles are supplementary)
Sum of the exterior angles of a polygon is 360°.

Let there be n sides of a polygon.

$$\begin{gathered} 72n=360 \\ n=\frac{360}{72} \\ n=5 \end{gathered}$$

Since it has 5 sides, the polygon is a pentagon.

(v) A pentagon has 5 diagonals.

$$\begin{gathered} \mathrm{If}n\mathrm{is}\mathrm{the}\mathrm{number}\mathrm{of}s\mathrm{ides,}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{diagonals}=\frac{n(n-3)}{2} \\ \frac{5(5-3)}{2} \\ =5 \end{gathered}$$