# Octagon

## Definition

An octagon is a polygon having eight sides and eight angles. It has eight vertices and eight edges that are joined end to end to form a close geometric shape.

An octagon-shape symbolizes rebirth, regeneration, transition, and infinity. The word ‘octagon’ is derived from the Greek words ‘okta’ meaning ‘eight’ and ‘gon’ meaning ‘side; angle’.

When all the eight sides and eight angles of an octagon are equal, it is a regular octagon. Otherwise, it is an irregular octagon.

The below-given properties and formulas to find the perimeter, area and length of diagonals only apply to the regular octagon.

## Properties

1. Has 8 sides of equal length; in octagon ABCDEFGH, AB = BC = CD = DE = EF = FG = GH = HA
2. Has 8 interior angles each measuring 135°; so ∠ABC = ∠BCD = ∠CDE = ∠DEF = ∠EFG =∠FGH = ∠GHA = ∠HAB
3. The sum of all eight interior angles is 1080°; so ∠ABC + ∠BCD + ∠CDE + ∠DEF + ∠EFG +∠FGH + ∠GHA + ∠HAB = 1080°
4. Has 8 exterior angles, each measuring 45°
5. Has 20 diagonals; example AC, BD, CE, DH, EH, and FH

## Formulas

### Perimeter

The formula for finding the perimeter of an octagon is given below:

Find the perimeter of a regular octagon having each side measuring 20 cm.

Solution:

As we know,
Perimeter (P) = 8a, here a = 20 cm
= 8 x 20 cm
= 160 cm

### Area

The formula for finding the area of an octagon is given below:

Area (A) = 2a2 (1+√2), here a = side

Find the area of a regular octagon having each side measuring 12 cm.

Solution:

As we know,
Area (A) = 2a2 (1+√2), here a = 12 cm
= 2 x (12)2 (1 +√2)
= 2 x 144 (1 +√2)
= 288 (1 +√2) = 289.41 cm

### Angles

#### Interior Angle

The angle formed inside the octagon at its corners when the line segments join in an end-to-end fashion.

Sum of Interior Angles

The total measure of all the interior angles combined in the octagon. The formula is given below:

Formula:

Sum of the interior angles = (n-2) x 180°, here n = number of sides

In octagon ABCDEFGH, n = 8

Thus,

Sum of the interior angles = (8 -2) x 180°

= 1080°

One Interior Angle

The measure of one interior angle can be obtained by dividing the sum of the interior angles with the number of sides in an octagon. The formula is given below:

Formula:

One interior angle = (n-2) x 180°/n, here n = number of sides

In octagon ABCDEFGH, n = 8

Thus,

Sum of the interior angles = (8 -2) x 180°/8

= 135°

#### Exterior Angle

The angle formed by any side of the octagon and the extension of its adjacent side. The formula is given below:

Formula:

Exterior angle = 360°/n, here n = number of sides

In octagon ABCDEFGH, n = 8

Thus,

Exterior angle = 360°/8

= 45°

### Length of the Diagonals

The line segments joining opposite corners in an octagon are its diagonal. In total, a regular octagon has 20 diagonals. There are three types of diagonals in them based on length, they are:

• Short diagonals such as EG, AC, and CE
• Medium diagonals such as CH, DG, and EH
• Long diagonals such as DH, AE, and CG

The formulas to determine their length is given below:

Find the length of the longest diagonal of a regular octagon having each side measuring 5.5 cm.

Solution:

As we know,
f = s√(4 + 2√2), here s = 5.5
= 5.5√(4 + 2√2)
= 14.37 cm

## Types

Depending on the sides, angles, and vertices, octagon shapes are classified into the following types:

1. Regular Octagon: Has eight sides of equal length and eight interior angles each measuring 135° and exterior angles of 45° each. The central angle also measures 45°. It has eight lines of symmetry and rotational equilibrium of order eight.
2. Irregular Octagon: Does not have all sides equal or all angles equal. Although all angles are not equal, but their sum is equal to 1080°.
3. Convex Octagon: Has all vertices pointing outwards or no angles are pointing inwards. No angles of a convex octagon measure more than 180°. A convex octagon can be both regular and irregular.
4. Concave Octagon: Have at least one vertex pointing inwards whose angle is greater than 180°. Thus all concave octagons are irregular.

## Real Life Examples

• The street stop sign.
• The Tower of the Winds in Athens, Greece is the oldest known octagonal building. Some other popular octagonal buildings are Dome of the Rock (Jerusalem, Isreal), Octagon (Roosevelt Island, in New York), Baptistery (Albenga, Italy), and Liaodi Pagoda (China) among many others.
• Objects in our daily life such as octagonal-shaped windows, floor tiles, lawn, clock, candles, and mirrors.
• More Resources

1. GreatWightSpark says: