Question

The sum of the interior angles of a triangle is 180°, the sum of those of a quadrilateral is 360, the sum of those of a pentagon is 540, and so on. Assuming that this pattern continues, find the sum of the interior angles a dodecagon (a 12-sided polygon)?

Answer 1

Sum of interior angles of a dodecagon is `1800^@`

Explanation:

Observe the pattern

No. of sides `{3,4,5,.........................}` each term rising by `1`. Note a dodecagon, a `12` sided polygon will be `10^(th)` term.

sum of interior angles `{180^@,360^@,540^@,........}` each term rising by `180^@` an arithmetic sequence, with first term as `180^@`

as `n^(th)` term of arithmetic sequence, whose first term is `a_1` and common difference is `d` is `a_n=a_1+(n-1)d`.

Hence, `10^(th)` term of the sequence representing sum of interior angles of a dodecagon is `180^@+(10-1)xx180^@=1800^`.

Note: Sum of interior angles of a polygon with `n` sides is `180^@xx(n-2)`.