Question

The sum of the measures of the interior angles of a convex polygon is \[{540^ \circ }\]. How do you classify the polygon by the number of sides?

Answer 1

Hint: If n are the number of sides of a convex polygon, the sum of its interior angles is \[\left( {n - 2} \right) \times {180^ \circ }\]. By using this formula we will find the value of $'n'$.
Also, the closed figure which is formed by five straight lines is known as a pentagon.

Complete step-by-step solution:
Sum of interior angles for \[n\] sides is
\[ \Rightarrow \left( {n - 2} \right) \times {180^ \circ } = {540^ \circ }\]
Shifting constants to other side we get,
\[ \Rightarrow \left( {n - 2} \right) = \dfrac{{{{540}^ \circ }}}{{{{180}^ \circ }}}\]
We divide the RHS (Right Hand side) term, we get
\[ \Rightarrow n - 2 = 3\]
We keep the variable in the left hand side and remaining term in the RHS, we get
\[ \Rightarrow n = 3 + 2\]
We just added the RHS terms,
\[ \Rightarrow n = 5\]

Since, the number of sides is \[5\], this polygon is a pentagon.

Note: Name of a polygon is termed according to the number of sides they have. Though a polygon can have an infinite number of sides and can be termed according to the number of sides, suppose a polygon consists. Suppose a polygon consists of 3 sides then it is known as a triangle, but if a polygon consists of \[234\] sides it will be known as \[234\]-gon.
Classified by Measuring the Angles-
Regular Polygon: Polygon in which all the sides of an angle are the same and all the interior angles of the polygon are equal is called regular polygon.
Irregular Polygon: The polygon in which it is not regular is known as Irregular Polygon. Irregular polygons do not have equal sides and measure of angles.
Convex Polygon: The polygon whose measure of an interior angle is less than 180° is termed as a convex polygon. The vertices of a convex polygon are always outwards. Concave Polygons: In concave polygons, at least one angle measures more than 180°.
Triangle: The three-sided polygon is known as the triangle.