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Area of Polygon in Coordinate Geometry


Polygon can be defined as packed or closed shape made by joining more than two line segments end to end. The sides of a Polygon never bisect each other. The Point where two lines meet is known as vertex of polygon.
Now we will see how to find the Area of Polygon in coordinate Geometry?

For finding the area of polygon we have to follow some steps which are given below:

Step 1: First we take two axis in a plane which is named as ‘x’ and y- axis.
Step 2: Then we draw a polygon on the plane.
Step 3: Then we find the coordinates of given polygon.
Step 4: If we have all the coordinates of a polygon then we can easily find the area of polygon by putting all the coordinates in formula shown below.

The formula for finding the area of polygon by using the coordinates:

Area of ploygon = $\frac{1}{2}$ $\begin{vmatrix} x_{1} & y_{1} \\ x_{2} & y_{2}\\ x_{3} & y_{3}\\ .& .\\ . & .\\ x_{n} & y_{n}\\ x_{1}& y_{1} \end{vmatrix}$ = $\frac{1}{2}$ | (x1 y2 - y1 x2) + (x2 y3 - y2 x3) + ....... + (xn y1 - yn x1) |
Now we see how to find the area of polygon by using the above steps:

A polygon is given then we can find the area of polygon by putting the values in the formula:

So we can write it as:

P 5 9
Q 8 7
R 10 3
S 3 3
P 5 9

Multiply 5 by 9, we get 45 and multiply 8 by 9 we get 72;

5 * 9 = 45;

8 * 9 = 72;

So we can write it as:

45 – 72 = -27;

When we multiply 8 by 3 we get 24 and multiply 10 by 7 we get 70;

8 * 3 = 24;

10 * 7 = 70;

So we can write it as:

24 – 70 = -46;

When we multiply 10 by 3 we get 30 and multiply 3 by 3 we get 9;

10 * 3 = 30;

3 * 3 = 9;

So we can write it as:

30 – 9 = 21;

When we multiply 3 by 9 we get 27 and multiply 5 by 3 we get 15;

3 * 9 = 27;

5 * 3 = 15;

So we can write it as:

27 – 15 = 12;

Now add all the value and then divide the number by 2 we get the area of polygon.

So the sum of all the values is:

=> -37 – 46 + 21 + 12;

=> -83 + 33;

=> -50;

Area = $\frac{1}{2}$ |-50| = $\frac{50}{2}$ = 25

Area of polygon is 25 square units.

Then the area of polygon is:

A = | -40/ 2|

A = 20 inch2

Area of a Regular Polygon

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A Polygon is a closed figure with all sides connected end to end. A polygon is a plane figure which is bounded by an enclosed path composite of a finite sequence of a straight line segments. These segments are the sides and the Point where they meet is polygon vertex or the corners.

There are many types of polygon and one of them is Regular Polygon. A regular polygon is a polygon whose all angles and all sides are equal. A convex and a star can be a regular polygon. Like all other polygons, the lines of a regular polygon do not bisect each other. The point where the lines of the polygon meet is known as vertex of polygon and area is the space occupied by the polygon.

Some of its properties are as follows:
A regular n - sided polygon has the property of Rotational Symmetry (those symmetry which seems to be same after its rotation) of the ‘n’ order.
All vertices of the Circle lie on the common co cyclic points. We can say that regular polygon follows the cyclic property.
All sides of regular polygons are equal.
Its examples are Equilateral Triangle, square, pentagon, hexagon, octagon, and decagon. These all are the simple regular polygon and are convex. Those regular polygons with the same number of sides are called similar polygon.
Formula for Area of a regular polygon is:
Area = (1/2)(apothem)(perimeter):
Where area stands for the area of regular polygon, apothem stands for the length of radius of the inscribed circle, and perimeter is the sum of all the sides of the regular polygon.

Let’s understand it in a simpler way with the help of an illustration. We will consider a Square to understand it in a simpler form. Consider a side of the square as 10 yards and its apothem as 5 yards, place the described data in the above mentioned formula the area of regular polygon (square) will be 1/2*5*40 = 100 yards square.

Area Of Polygon Given the length of a side

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A Polygon is a geometrical figure on a plane formed with straight line segments. Let us see how we can use the length of side to find other sides and all other perimeters of the polygons.

Let us see some of these geometrical figure and shapes.

1. Area of polygon

To find the area of polygon having Coordinate (x1 , y1), (x2 , y2) , (x3 , y3) .... (xn , yn) Plane we use the following formula:

Area of Polygon = $\frac{1}{2}$ | (x1 y2 - y1 x2) + (x2 y3 - y2 x3) + ....... + (xn y1 - yn x1) |

When sides of Regular Polygons are given we can find the area of the polygons by using particular formula:

1. Square

If we have given the length of a side of a Square, we can find:

1. Area of Square

As all sides of a square are equal we can find area of the square by squaring one of the side i.e. (side)2

2. Perimeter of Square = 4(a)

The perimeter of a square is the total distance covered by sides of the square. It is the path that surrounds area of the square. The “perimeter” word is a Greek word in which "Peri" means "around" and the "meter" means “to measure". To find the perimeter of the square we simply add the length of each side together

As the sides of the square are equal in length therefore; Perimeter = 4* (side).

3. Diagonal of Square

To find the diagonal of square we simply multiply its side by the square root of 2.

Diagonal of Square = (side) * (sqrt (2))

1. Right angled triangle

Using Pythagorean Theorem, we can find the length of sides of the triangle using the following formula:

(Hypotenuse) 2 = (Base) 2 + (Perpendicular) 2

1. Area of a triangle

When we have given the lengths of the triangle we can find area of the triangle using Heron’s Formula:

According to Heron’s Formula:
Area of the triangle = √s (s-a) (s-b) (s-c)

Where a, b, and c are the sides of the triangle
And ‘s’ is the perimeter of the triangle.

s = $\frac{a+b+c}{2}$

Algorithm to Find the Area of Polygon

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Polygon can be defined as a shape which is comprised of several line segments which are connected together to form a closed figure. In other words, a Polygon is limited by a closed path. Line segments in a polygon can be of different or same length. The line segments are known as edges or sides of polygon. Verticex is the Point where two edges or sides meet. If sides in a polygon are three then it is known as triangle, if it is five then pentagon and if seven then heptagon and so on.

If the coordinates of vertices of a polygon are known then an algorithm can be used to find the area of polygon. Here ‘a’ and ‘b’ are the parameters which can be called as arrays of the nodes (vertices). These arrays are drawn in clockwise direction beginning from any of the vertex. If rotation of vertices is taken as anticlockwise (counterclockwise – direction of rotation is opposite to that of the watch) then an subtraction sign (- sign) will be placed in final result. The vertices (nodes) are numbered as shown in the figure. Define the Numbers by nump in the algorithm. Using these identities, we can calculate the area of a polygon. Let’s see the Algorithm to find the area of Polygon. Let’s consider some variables as ‘s’ denoting area, nump denotes the numbered given to the node. This can be given as follows:

Function polygon Area (a, b, nump)
s = 0; // accumulating surface area in the loop
p = nump – 1: //the final node (vertex) is the previous one to the first
for (q=0; q < nump ; q++)
s = s + (a[p] + a[q]) * (b[p] – b[q]);
p = q; //p is previous node to q
return s / 2.