Area Of Any Polygon Formula: A Complete Guide

Mei 23, 2023 Posting Komentar

Area of Regular Polygon Calculator from www.math-salamanders.com

Calculating the area of a polygon is an essential skill for anyone working in fields such as engineering, architecture, or even art. The formula for finding the area of a polygon varies depending on the type of polygon. However, in this article, we will explore the area of any polygon formula, which can be applied to any polygon, regardless of its shape or size.

What is a Polygon?

A polygon is a closed shape that has three or more straight sides. The sides of a polygon do not intersect each other, and the angles between them are always less than 180 degrees. Some common examples of polygons are triangles, rectangles, and hexagons.

How to Find the Area of a Polygon?

The formula for finding the area of a polygon is:

Area = 1/2 x perimeter x apothem

Where:

Perimeter is the sum of the lengths of all the sides of the polygon.
Apothem is the distance from the center of the polygon to the midpoint of any side.

Step-by-Step Guide

Let's take a look at a step-by-step guide to finding the area of a polygon using the formula:

Measure the length of each side of the polygon.
Add up the lengths of all the sides to find the perimeter.
Measure the distance from the center of the polygon to the midpoint of any side to find the apothem.
Substitute the values of the perimeter and apothem into the formula.
Multiply the result by 1/2 to find the area of the polygon.

Examples of Finding the Area of a Polygon

Let's take a look at some examples to help you understand how to apply the formula:

Example 1: Regular Hexagon

Let's say we have a regular hexagon with a side length of 5 cm. To find the area of the hexagon, we can follow these steps:

Since the hexagon has six sides, each with a length of 5 cm, the perimeter is 6 x 5 = 30 cm.
The apothem of a regular hexagon is equal to the radius of the circle inscribed inside it. The radius can be found using the formula r = (s/2) x √3, where s is the length of a side. So, in this case, the apothem is (5/2) x √3 = 4.33 cm.
Substituting the values into the formula, we get:

Area = 1/2 x 30 x 4.33 = 64.95 cm^2

Example 2: Irregular Pentagon

Let's say we have an irregular pentagon with side lengths of 6 cm, 5 cm, 8 cm, 7 cm, and 9 cm. To find the area of the pentagon, we can follow these steps:

The perimeter is the sum of the lengths of all the sides, which is 6 + 5 + 8 + 7 + 9 = 35 cm.
Since the pentagon is irregular, we cannot use the formula for the apothem. Instead, we can divide the pentagon into triangles and use the formula for the area of a triangle. In this case, we can divide the pentagon into three triangles, as shown below:

Pentagon divided into triangles

To find the area of each triangle, we can use the formula:

Area of Triangle = 1/2 x base x height

For the first triangle, the base is 6 cm, and the height is 4 cm. So, the area is:

Area of Triangle 1 = 1/2 x 6 x 4 = 12 cm^2

Similarly, for the second triangle, the base is 5 cm, and the height is 4 cm. So, the area is:

Area of Triangle 2 = 1/2 x 5 x 4 = 10 cm^2

For the third triangle, the base is 7 cm, and the height is 3 cm. So, the area is:

Area of Triangle 3 = 1/2 x 7 x 3 = 10.5 cm^2

The total area of the pentagon is the sum of the areas of the three triangles:

Area = 12 + 10 + 10.5 = 32.5 cm^2

Conclusion

Calculating the area of a polygon is a simple process, provided you know the formula. By following the steps outlined in this article, you can find the area of any polygon, regardless of its shape or size. Remember, practice makes perfect, so keep practicing until you master this essential skill.

Happy calculating!