E Board: Suggested Books     ICSE Board:  Foundation Mathematics 
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In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit. Or simply a closed plane figure bounded by three or more line segment to form a closed loop is called a polygon.p1

Nomenclature:

  • The line segments forming the polygon are called sides.
  • The point of intersection of two line segments is called a vertex.
  • Number of vertices of a polygon is equal to the number of line segments or sides.

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Different types of Polygons:  This is based on the number of sides that the polygon has. Here are few examples:P10

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p4Diagonal of a Polygon: A line segment joining any two non-consecutive vertices is called

a diagonal of the polygon.

The dotted lines are diagonals of the shown polygons.

 

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Interior and Exterior Angles of a Polygon: This is an important concept.

\angle 1 \ and\  \angle 2  are interior angles. These are made by the two sides of the polygon.p5

\angle 3  is called an exterior angle. This is formed by extending a side of the polygon as shown in the adjacent figure.

Just by looking at the figure you can tell that

\angle 2  +   \angle 3 = 180^{\circ} 

Hence we can say that:

Exterior Angle + Adjacent Interior Angle = 180^{\circ} 

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Convex Polygon vs Concave Polygonp6

If the interior angle of the polygon is less than = 180^{\circ} , then it is called convex polygon. If you look at any of the polygons shown above, you will see that all the interior angles are less than = 180^{\circ} .

But there can be cases where the interior angle of a polygon could be more than = 180^{\circ} . Take a look at the adjacent figure. Here you will see that \angle 1 > 180^{\circ}  (which is a reflex angle).

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Regular Polygon: A polygon that satisfies the following condition is called a regular polygon.m19x

  1. All sides are equal
  2. All interior angles are equal
  3. All exterior angles are equal

 

For a regular polygon with $latex n &s=1$ sides we have the following:

Each Interior Angle = [\frac{(2n-4)\times 90^{\circ}}{n}]^{\circ}

 

Proof:

A polygon can be divided into (n-2) triangles.

See a few examples in the adjacent figure.l52

We know that the sum of the angles of a triangle is = 180^{\circ} .

Therefore the sum of the interior angles = (n-2) \times 180 = (2n - 4) \times 90^{\circ}

\Rightarrow Interior \ Angle = \frac{(2n-4)\times 90^{\circ}}{n}

Each \  Exterior \ Angle = [\frac{360}{n}]^{\circ}

 

 

 

Proof:

Sum of all Interior Angles + Sum of all Exterior Angles = n \times 180 = n \times (2 \times 90)^{\circ}

Sum of all Exterior Angles = n \times (2 \times 90)^{\circ} -  (2n - 4) \times 90^{\circ}=360^{\circ}

Each \  Exterior \ Angle = [\frac{360}{n}]^{\circ}

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m18x

3. From the above point 2, we can also say that

n = [\frac{360}{Each \ Exterior \ Angle}]^{\circ}

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4. We already know that

Exterior Angle + Adjacent Interior Angle  = 180^{\circ} 

\Rightarrow \ Exterior \ Angle=180^{\circ} - Adjacent \ Interior \ Angle

 

E Board: Suggested Books     ICSE Board:  Foundation Mathematics 
Class 8: Reference Books     Class 8: NTSE Preparation 
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