Question

The interior angles of an octagon are in \[{\text{A.P.}}\]. The smallest angle is \[30^\circ \] and the common difference is \[30^\circ \]. Find the sum of all the angles.

Answer 1

Hint: Here, we will use the formula for sum of A.P. is \[{{\text{S}}_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\], where \[a\] is the smallest angle, \[d\] is the common difference and \[n\] is the number of angles. Then, substitute the values of \[a\], \[d\] and \[n\] in expression for \[{{\text{S}}_n}\].

Complete step-by-step solution:
Given that an octagon has 8 sides.
We will use the formula for sum of all angles \[{{\text{S}}_n}\] is \[\dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\], where \[a\] is the smallest angle, \[d\] is the common difference and \[n\] is the number of angles.

Since it is given that the interior angles of an octagon are in A.P, the angles can be written as \[a\], \[a + d\], \[a + 3d\], \[a + 4d\], \[a + 5d\], \[a + 6d\] and \[a + 7d\].

Now we will find the values of \[a\], \[d\] and \[n\].

\[a = 30\]
\[d = 30\]
\[n = 8\]

Substituting these values of \[a\], \[d\] and \[n\] in expression for \[{{\text{S}}_n}\], we get

\[
  {{\text{S}}_n} = \dfrac{8}{2}\left[ {2\left( {30} \right) + \left( {8 - 1} \right)30} \right] \\
   = 4\left[ {60 + 7\left( {30} \right)} \right] \\
   = 4\left[ {60 + 210} \right] \\
   = 4\left[ {270} \right] \\
   = 1080 \\
\]

Thus, the sum of all angles of an octagon is \[1080^\circ \].

Note: In this question, some students mistakenly write the formula for the sum of all angles. Also, we are supposed to write the values properly to avoid any miscalculation.