\displaystyle \textbf{1. Polygon: }\text{A closed plane figure bounded by three or more line segments }
\displaystyle \text{is called a polygon. These line segments are called its sides, and the line segment}
\displaystyle \text{ joining two non-consecutive vertices of a polygon is called its diagonal. The point of }
\displaystyle \text{intersection of two consecutive sides of a polygon is called a vertex.}
\displaystyle \\

\displaystyle \textbf{2. Convex Polygon: }\text{If each angle of a polygon is less than }180^\circ,\text{ then it is called}
\displaystyle \text{a convex polygon.}
\displaystyle \\

\displaystyle \textbf{3. Concave Polygon: }\text{If at least one angle of a polygon is a reflex angle, i.e., more than}
\displaystyle 180^\circ,\text{ then it is called a concave polygon.}
\displaystyle \\

\displaystyle \textbf{4. Regular Polygon: }\text{If all sides of a polygon are equal and all angles are equal, then it}
\displaystyle \text{is called a regular polygon.}
\displaystyle \\

\displaystyle \textbf{5. Theorems:}
\displaystyle \text{(i) The sum of all the interior angles of a convex polygon of }n\text{ sides is}
\displaystyle (2n-4)\text{ right angles.}
\displaystyle \text{(ii) The sum of all the exterior angles of a convex polygon is }4\text{ right angles.}
\displaystyle \\

\displaystyle \textbf{6. Some More Results:}
\displaystyle \textbf{(i) For All Convex Polygons:}
\displaystyle \text{(i) Sum of all interior angles of a polygon of }n\text{ sides}=(2n-4)\text{ right angles.}
\displaystyle \text{(ii) Sum of all exterior angles of a polygon of }n\text{ sides}=4\text{ right angles.}
\displaystyle \text{(iii) At each vertex of a polygon, we have:}
\displaystyle \text{Interior Angle}+\text{Exterior Angle}=180^\circ.
\displaystyle \\

\displaystyle \textbf{(ii) For Regular Polygons:}
\displaystyle \text{(i) Each interior angle of a regular polygon of }n\text{ sides}
\displaystyle =\frac{2n-4}{n}\text{ right angles}
\displaystyle =\left[\frac{(2n-4)\times90^\circ}{n}\right].
\displaystyle \text{(ii) Each exterior angle of a regular polygon of }n\text{ sides}=\left(\frac{360}{n}\right)^\circ.
\displaystyle \text{(iii) If each exterior angle of a regular polygon is }x^\circ,\text{ then the number of its sides}
\displaystyle =\left(\frac{360}{x}\right).
\displaystyle \textbf{Note: }\text{Greater is the number of sides in a regular polygon, greater is the value of its}
\displaystyle \text{interior angle and smaller is the value of each exterior angle.}
\displaystyle \\

\displaystyle \textbf{(iii) Number of diagonals in a polygon of }n\textbf{ sides}
\displaystyle =\left[\frac{n(n-1)}{2}-n\right]
\displaystyle =\frac{n(n-3)}{2}.

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.