Other Polygons

We have other types of polygons aside from triangles and quadrilaterals.

Examples are

Polygon	Number of sides
Pentagon	5 sides
Hexagon	6 sides
Heptagon	7 sides
Octagon	8 sides
Nonagon	9 sides
Decagon	10 sides

Types of Polygons:

Regular Polygon: A regular polygon is a polygon with its sides and angles equal.
Irregular Polygon: These are polygons with unequal lengths of sides and angles.
Convex: A convex polygon has all its interior angles either acute or obtuse i.e less than 180°.
Concave Polygon: In a concave polygon at least one of its interior angles is reflex.

Interior Angles of a Polygon:

The interior angles of a polygon are the angles inside the polygon. A polygon has the same number of sides as the number of angles inside it.

Formulae:

1. Sum of interior angles = (2n – 4) × 90°

=  (n – 2) × 180°

Where n = number of sides

e.g. for a pentagon,  n = 5

sum of interior angles of a pentagon = [(2 × 5) – 4] × 90°

= (10 – 4) × 90

= 6 × 90

= 540°

2. The sum of interior angles can also be obtained from the number of triangles in the polygon.

Sum of interior angles of pentagon = 180° × 3 = 540°. Because it has 3 triangles, and the sum of angles in a triangle is 180°.

Sum of interior angles of hexagon = 180 × 4 = 720^°

Heptagon – 5 triangles
Octagon – 6 triangles
Nonagon – 7 triangles
Decagon – 8 triangles

3. Size of each interior angle = \( \frac{sum\:of\:interior\:angles}{numer\:of\:sides}\\ = \frac{(2n\:-\:4)\:\times \:90}{n} \)

Example 10.4.1:

a. Calculate the total internal angles of an octagon.
b. Calculate the size of each angle of a regular octagon.

Solution

a. For octagons, n = 8

Sum of interior angles = (2n – 4) × 90°

   = ((2 × 8) – 4) × 90°

    = (16 – 4) × 90°

    = 12 × 90° = 1080°

b. Size of each interior angle = \( \frac{sum \: of \: int \: angle}{n} \\ =\frac{1080}{8} \\ = \scriptsize 135^o \)

Example 10.4.2:

The interior angles of a hexagon are 100°, 90°, 110°, 120°, 95° and y°, calculate y.

Solution

n = 6

Sum of interior angle = (2n – 4) × 90°

100° + 90° + 110° +  120° + 95° + y° = ((2 × 6) – 4) × 90°

515° + y° = (12 – 4) × 90°

515° + y° = 720°

y = 720° – 515°

y = 205°

Exterior Angles of a Polygon:

1. The sun of exterior angles of a polygon = 360°
x + y + z + r + w = 360°

2. Size of each exterior angle = \( \frac{360^{\circ}}{n} \)

3. Exterior angle + corresponding interior angle = 180°

a + b = 180° (Sum adjacent angles in a straight line)

Example 10.4.3:

Calculate:

a. The exterior angle.
b. The number of sides

of a regular polygon with an interior angle of 60°.

Solution

a. ext angle  + Int. angle = 180°

ext angle + 60° = 180°

ext. angle = 180° – 60°

ext. angle = 120°

b. Each external angle = \(\frac{360}{n}\)

n = \(\frac{360}{each \: ext.angle}\\ = \frac{360}{12} \\ \scriptsize 3 \: sides\)

Example 10.4.4:

The exterior angles of a hexagon are shown below. Find the value of x

Solution

Sum of the exterior angles = 360°

(2x + 4) + (3x – 8) + (4x – 13) + (5x – 3) + (x + 2) + (2x + 4) = 360°

2x + 4 + 3x – 8 + 4x – 13 + 5x – 3 + x + 2 + 2x + 4 = 360°

collect like terms

2x + 3x + 4x + 5x + x + 2x + 4 – 8 – 13 – 3 + 2 + 4 = 360°

17x – 14 = 360

17x = 360 + 14

17x = 374

divide both sides by 17

\( \frac{17x}{17} = \frac{374}{17}\)

\( \scriptsize x = 22^{\circ}\)