Page 180 - ICSE Math 8

Diagonals of a polygon diagonals
The line segments obtained by joining non-adjacent vertices are called diagonals of the polygon.
D and D are the two diagonals in Fig. 16.8. D 2
2
1
Only one diagonal can be drawn from any vertex of a polygon, except for a triangle. A hexagon D 1
can have three diagonals emerging from any vertex. Thus, we see that if a polygon has n sides,
then (n – 3) diagonals can be drawn from any one vertex of a polygon. Also, (n – 2) triangles Fig. 16.8
are formed as a result.

Sum of Interior Angles of a Polygon
You know that if a polygon has n sides, then (n – 3) diagonals can be drawn from any one vertex, which divides
the polygon in (n – 2) triangles.
Thus, if a polygon has n sides, there will be (n – 2) triangles formed.
We know that, sum of angles of a triangles = 180°
∴ Sum of the angles of (n –2) triangles = (n – 2) × 180°
or, Sum of angles (interior angles) of a polygon with n sides = (2n – 4) × 90° or (2n – 4) right angles

Sum of Exterior Angles of a Polygon E 4 D

In Fig. 16.9, if we start moving forward from point A to B and then to C, 5 3
D, E and return to A, we have moved by one complete angle, i.e., 360º. C
Thus, –1 + –2 + –3 + –4 + –5 = 360º A 2

The sum of measure of exterior angles of any polygon is 360º. 1 B
Fig. 16.9
Regular and Irregular Polygons
In a regular polygon (Fig. 16.10), all sides are of the same length
(equilateral) and all angles are of the same measure (equiangular).
A polygon which is not regular is an irregular polygon.
Thus, in a regular polygon of n sides, we have
Sum of the interior angles = (2n – 4) × 90° (a) Fig. 16.10 (b)
As all the angles are equal in a regular polygon, we have
(2n – 4) × 90° Maths Info
Each interior angle =
n
Also, sum of exterior angles = 360° • All regular polygones are convex
360 ° polygons.
Each exterior angle = • At each vertex of any polygon,
n
Number of sides of a regular polygon = 360° exterior angle + interior angle = 180°.
Each exterior angle
Quadrilaterals B
A four-sided polygon is called a quadrilateral (Fig. 16.11). It has: A
• four sides (AB, BC, CD, DA)
• four vertices (A, B, C, D)
• four angles (–DAB, –ABC, –BCD, –CDA) D C
• two diagonals (AC and BD) Fig. 16.11
The sides having a common vertex are known as adjacent sides. For
example, sides AB and BC, BC and CD, CD and DA, DA and AB are Maths Info
adjacent sides.
The sides with no common vertex are known as opposite sides. For A quadrilateral has infinite number of
interior and exterior points.
example, sides AB and DC, AD and BC are opposite sides.

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