How Many Sides Does a Polygon Have to Have to Be Considered an Octagon?
A polygon is a two-dimensional shape with straight sides. A polygon must have at least three sides. The naming of polygons follows a specific pattern, except for triangle and quadrilateral. For polygons with more than four sides, the number of sides is prefixed with Greek or Latin derivatives. An octagon, for instance, has 8 sides, derived from the Greek word 'octa' meaning 'eight.' Other polygons include polygons with 5, 6, 7, 9, 10, 11, and 12 sides, known as pentagon, hexagon, heptagon, nonagon, decagon, undecagon, and dodecagon respectively.
Understanding Octagons
Let's delve deeper into the characteristics of an octagon. The prefix 'octa' means 'eight,' and an octagon is a polygon with 8 sides. Similarly, the prefix meanings for other polygons are as follows: 5 sides - pentagon, 6 sides - hexagon, 7 sides - heptagon, 9 sides - nonagon, 10 sides - decagon, 11 sides - undecagon, and 12 sides - dodecagon.
Types of Octagons
Octagons can be broadly categorized into two types: regular octagons (RO) and irregular octagons (IO).
Regular Octagons (RO)
They have eight sides. All 8 sides are equal in length. The octagon is always a convex figure. Joining the opposite vertices forms 8 identical congruent isosceles triangles. The sum of the internal angles at the periphery is 1080 degrees, with each angle being 135 degrees. Each of the 8 external angles is 45 degrees, and their sum is 360 degrees. Joining opposite angles or midpoints of opposite sides results in an axis of symmetry, with the figure being the mirror image of the other half. Each of the 8 isosceles triangles has base angles of 67.5 degrees and the third angle is 45 degrees. The intersection of the diagonals joining opposite vertices results in a common point, serving as the circumcentre of a circle passing through all 8 vertices. The radius of the circumcircle is equal to the length of the equal sides of the isosceles triangles. The diagonals connecting alternate vertices form two overlapping squares. The number of diagonals is given by the formula ( frac{n(n-3)}{2} 20, ) where ( n 8 ).Irregular Octagons (IO)
They have eight sides. The sides are not equal in length. The octagon can be either convex or concave. Joining the opposite vertices forms 8 scalene triangles. The sum of the internal angles at the periphery is 1080 degrees, but the angles are not equal. Each of the 8 external angles is not 45 degrees or equal, summing up to 360 degrees. Joining opposite angles or midpoints of opposite sides results in an axis of symmetry, but the figure is not the mirror image of the other half. The triangles formed are not isosceles or equal, with different angles. Locating the circumcentre is not easy and may be impossible. The radius of the circumcircle is not equal to the length of the sides. The diagonals connecting alternate vertices form two irregular quadrilaterals. The number of diagonals is still given by the formula ( frac{n(n-3)}{2} 20, ) where ( n 8 ).Understanding the characteristics of a polygon, specifically an octagon, is crucial in many fields such as architecture, engineering, and design. Whether regular or irregular, the properties of an octagon provide a rich field of study and application.
By exploring the different types and properties of these geometric shapes, we can gain a deeper appreciation for the beauty and complexity of mathematics.