Polygon Area Formula

When dealing with polygon area problems in mathematics and real-world scenarios, understanding the definition of polygons and their area calculation formula is essential. Join Mytour on this journey to unravel the mysteries of polygons and master the art of area calculation through clever tips.

Calculating the Area of Polygons: Regular and Irregular

I. Definition, Classification of Polygons

1. What is a Polygon?

A polygon is a closed planar figure, formed by consecutive line segments lying on the same plane and creating a closed loop.

- A flat section defined by the zigzag lines of a polygon is known as the polygonal area.

- A straight segment on the zigzag line of a polygon is referred to as the edges of the polygon.

- The connecting point between two edges is called the vertex of the polygon.

- A straight segment connecting two non-adjacent vertices of a polygon is known as the diagonal.
 

2. Polygon Classification

Polygons can have line segments (the edges of the polygon) and extremely large angles. Polygons are classified as follows:

- Convex Polygon: The entire polygon lies on one side of the line containing any edge of the polygon.

- Concave Polygon: The polygon lies on both sides of at least one line containing an edge of the polygon.

- Simple Polygon: Edges can only intersect at the vertices of the polygon, with no two non-adjacent edges intersecting.

- Non-Simple Polygon: A polygon with two non-adjacent edges intersecting (the intersection point is not a vertex of the polygon).

- Regular Polygon: All edges and all angles of the polygon are equal. Regular polygons are common in mathematics: equilateral triangle, square, regular pentagon, regular hexagon, etc.

II. Formula for calculating the area of any polygon with examples

1. Formula for calculating the area of a regular polygon

A = (a x p)/2

Where:

- p is the perimeter of the polygon (perimeter is the sum of the lengths of its edges).

- a is the length of the apothem (apothem is a line segment perpendicular to an edge and passing through the center of the polygon).

Example of calculating the area of a regular polygon: A hexagon with 6 sides (n) where each side (s) measures 10cm. What is the area of this regular polygon?

Solution:

Step 1: Calculate the perimeter of the hexagon.

p = 6 x 10 = 60cm.

Step 2: Calculate the apothem

A hexagon has 6 sides (n), each side measuring 10 (s). We assign values 6 and 10 to n and s.

Apply the expression:

=> The result of the expression 2tan(180/6) = 1.1547.

Continuing, divide 10 by 1.1547, the obtained apothem 'a' is 8.66.

Step 3: Calculate the area of the regular polygon

Area = (a x p)/2 = (8.66 x 60)/2 = 259.8 (cm2).

2. Method for calculating the area of an irregular polygon

Step 1: Divide the polygon into small parts such as triangles, squares, rhombuses, parallelograms...

Step 2: Calculate the area of the small polygons.

Step 3: Calculate the area of the large polygon by adding the total area of the small polygons.

(*) Common method for calculating polygon area:

- Triangle Area Formula: A = 1/2 x height x length of the opposite side of the vertex.

- Square Area Formula: A = Side x Side.

- Parallelogram Area Formula: A = Height x corresponding base.

- Rectangle Area Formula: A = Length x Width.

Example problem on calculating the area of an irregular polygon: For an irregular polygon with sides: AB = ED = 20 cm, BC = CD = 5 cm, and AB = BD = 8 cm. Find the area of the polygon.

Solution:

III. Memorizing polygon area formulas effectively through poetry
 

* Trapezoid Area Formula:

To find the trapezoid area
Add the big base to the small one, don't ignore
Then multiply by the height, don't forget to do so
Divide the result by two, that's the way to go

* Rectangle Area Formula:

For a rectangle as you've been taught
Multiply the length by the width, the answer sought

* Calculate the area of a square - Formula for calculating the area of a regular polygon:

The square is indeed quite neat
Multiply the side by itself, the answer is complete

* Triangle Area Formula:

Triangles might seem a bit tough
Multiply the height by the base then halve, that's enough

* Circle Area Formula:

Finding the circle's area is quite a breeze
Multiply the radius, then by pi with ease

* Parallelogram Area Formula:

Parallelogram's area won't deceive
Multiply the height by the base, it's easy to believe.

With math formulas in rhymes, students can grasp how to calculate the area of regular and irregular polygons with ease. This lays the foundation for them to compute the area of a rectangle, square, or even a triangle. It's crucial for them to memorize these well. If students have any effective geometry learning methods or interesting math formulas, feel free to share with us.

Additionally, students should also understand the formulas for calculating the perimeter of polygons, as it is one of the fundamental and essential formulas they need to master.

Each trapezoid type has a different calculation formula, and ways to find the area of a trapezoid, isosceles trapezoid, or right trapezoid will be shared in the upcoming articles on Mytour.