Polygons
A closed
plane figure formed by three or more line segments is called a polygon. The word polygon is made of two words 'poly' and 'gon'. 'Poly' means 'many' and 'gon' means 'sides'. Thus, polygon means figure with many sides. Polygons are made with only line segments.
Types of Polygons
When each
interior angle of a polygon has a measure that is less than 180°, the polygon
is called a convex polygon. If one or more interior angles of a polygon are
reflex angles, it is called a non-convex polygon. In this section, we shall
discuss the properties of convex polygons.
A polygon is
named according to the number of sides it has. Here are some common polygons.
A polygon
with n sides can be called an n-gon. For instance, a polygon with 17 sides is
called a 17-gon.
A regular
polygon has equal sides and equal angles. The figures given below show a
regular pentagon and hexagon respectively.
Sum of Interior Angles of a Polygon
All the
angles inside the polygon are called its interior angles. We have learned that
the angle sum of a triangle is 180°.
“The sum of
the interior angles of an n-sided polygon is (n – 2) × 180°.”
If n = 3,
then the sum of the interior angles = (3 - 2) × 180° = 180°
If n = 4,
then the sum of the interior angles = (4 - 2) × 180° = 360°
Example 1: In the given figure, find the value
of angle x.
∴ ∠x = 117°
Example 2:
The angle sum of the
interior angles of an n-sided polygon is 3600°. Find the value of n.
Solution: Sum of the interior angles of an
n-gon = (n – 2) × 180° (∠ sum of
polygon)
∴ (n – 2) × 180° = 3600°
n – 2 = 3600/180
n – 2 = 20
n = 22
Sum of Exterior Angles of a Polygon
A polygon
has as many exterior angles as its number of sides. For example, if we produce
all the sides of a quadrilateral, then four exterior angles are formed. And if
we produce all the sides of a pentagon, then five exterior angles are formed.
Example 3:
Find the measure of
each exterior angle of a regular decagon.
Solution: Sum of the exterior angles of a
decagon = 360° (exterior ∠ sum of polygon)
∴ the measure of each exterior angle
of a decagon = 360° ÷ 10 = 36°
Example 4:
In the given figure,
find the measures of angles x and y.
Solution: ∠x + 89° = 180° (adj. ∠s on a st. line)
∠x = 91°
102° + ∠x + ∠y + 103° = 360° (ext. ∠ sum of polygon)
102° + 91° +
∠y + 103° = 360°
∴ ∠y = 64°