Question

What is the sum of the interior angle measures of a convex nonagon?

Answer 1

Hint: We are given a question asking us to find the sum of the interior angle of a convex nonagon. A nonagon refers to a polygon with nine sides. When we say a convex polygon, we refer to a polygon which has the measure of an interior angle of the polygon less than \[{{180}^{\circ }}\]. We know that, for a ‘n’ sided polygon, the sum of the interior angle is \[(n-2){{180}^{\circ }}\]. Here, substituting the value of the \[n=9\] in the above expression and solving the expression further, we will have the measure of the sum of the interior angle of a convex nonagon.

Complete step by step answer:
According to the given question, we are asked to find the measure of the sum of the interior angle of a convex nonagon.
A nonagon refers to a polygon which has a number of sides as nine. When we introduce a term like ‘convex’ before a polygon it simply means that the measure of the interior angle of that particular polygon is less than that of the straight line angle which is, \[{{180}^{\circ }}\].
We know that for a fact that the for a ‘n’ sided polygon, we have the sum of that particular polygon given by the following expression:
\[(n-2){{180}^{\circ }}\]
Where ‘n’ is the number of sides
Now, we will substitute the value of \[n=9\] in the above expression and we get the value of the expression as,
\[\Rightarrow (9-2){{180}^{\circ }}\]
Subtracting the terms within the brackets, we get,
\[\Rightarrow (7){{180}^{\circ }}\]
Multiplying the terms in the above expression, we get,
\[\Rightarrow {{1260}^{\circ }}\]

Therefore, the sum of the interior angle of a convex nonagon is \[{{1260}^{\circ }}\].

Note: The expression \[(n-2){{180}^{\circ }}\] is taken because for a polygon with ‘n’ sides, if we join one vertex to all other vertices, we will have triangles formed out of this construction and the number of triangles formed is given by \[(n-2)\]. As we know that the sum of all the angles of a triangle is equal to \[{{180}^{\circ }}\]. So, for a polygon with ‘n’ sides and having \[(n-2)\] triangles formed, we will have the sum of the interior angle as \[(n-2){{180}^{\circ }}\].