In geometry, a polygon is a shape made up of straight lines connected in a closed loop. It also has vertices equal to the number of sides. The following two geometric objects are polygons.
Definition of a regular polygon
If the sides of the polygon are the same size and the angles are also the same, then the polygon is called a regular polygon. Regular polygons follow.
The name of the polygons ends with the suffix “gon” and the number of sides determines the front part of the name. The number in Greek is used as a prefix, and the whole word says that it is a polygon with so many sides. A few examples follow, but the list goes on.
How to Find the Area of Polygons: Method
The area of a general irregular polygon cannot be determined directly from the formula. However, we can break the polygon into smaller polygons that we can use to easily calculate the area. The sum of these components then gives the area of the entire polygon. Consider an irregular heptagon as shown below. The area of the heptagon can be given as the sum of the individual triangles within the heptagon. By calculating the area of the triangles (a1 to a4).
Total area = a1 + a2 + a3 + a4
If there are more pages, more triangles have to be added, but the basic principle remains the same.
With this concept we can get a result for calculating the area of the regular polygons.
Consider the regular hexagon with length d sides as shown below. The hexagon can be broken down into six smaller congruent triangles, and these triangles can be rearranged by a parallelogram as shown.
The diagram shows that the sums of the areas of the smaller triangles are equal to the area of the parallelogram (rhombus). Therefore, we can determine the area of the hexagon from the area of the parallelogram (rhombus).
Area of the parallelogram = sum of the areas of the triangles = area of the heptagon
If we write an expression for the area of the diamond, we have
Area rhom = 3 ie
By rearranging the terms
From the geometry of the hexagon we can see that 6d is the perimeter of the hexagon and h is the perpendicular distance from the center of the hexagon to the perimeter. Hence we can say
Area of the hexagon = 12 circumference of the hexagon × perpendicular distance to the circumference.
From the geometry we can show that the result can be extended to polygons with any number of sides. Hence, we can generalize the above expression into
Area of the polygon = 12 perimeter of the polygon × perpendicular distance to the perimeter
The perpendicular distance to the perimeter from the center is called the apotheme (h). So if a polygon with n sides has a perimeter p and an apotheme h, we get the formula:
How to find the area of regular polygons: Example
- An octagon has a side length of 4 cm. Find the area of the Octagon. To find the area of the octagon, two things are required. That is the scope and the apothem.
- Find the perimeter
The length of one side is 4 cm and an octagon has 8 sides. Hence pperimeter of the octagon = 4 × 8 = 32cm
- Find the apotheme.
The interior angles of the octagon are 1350 and the side of the drawn triangle bisects the angle. Therefore we can calculate the apotheme (h) using trigonometry.
h = 2tan67.5 0 = 4.828cm
- Hence the area of the octagon