Question

We know that the sum of the interior angles of a triangle is 180°. Show that the sum of the interior angles of polygons with 3, 4, 5, 6, …. sides form an arithmetic progression. Find the sum of the interior angles for a 21 - sided polygon.

Answer 1

Show that: the sum of the interior angles of polygons with 3, 4, 5, 6, …. sides form an arithmetic progression.

To Find: The sum of the interior angles for a 21 - sided polygon.

Given: That the sum of the interior angles of a triangle is 180°.

NOTE: We know that sum of interior angles of a polygon of side n is (n – 2) x 180°.

Let a_n = (n – 2) x 180°

⇒ Since a_n is linear in n.

So it forms AP with 3, 4, 5, 6,……sides

{a_n is the sum of interior angles of a polygon of side n}

By using the above formula, we have

a₂₁ = (21 – 2) x 180°

a₂₁ = 3420°

So, the Sum of the interior angles for a 21 - sided polygon is equal to 3420°.