CSES Solution - Polygon Area

Your task is to calculate the area of a given polygon.

The polygon consists of n vertices (x₁,y₁),(x₂,y₂),.....(x_n,y_n). The vertices (x_i,y_i) and (x_i+1,y_i+1) are adjacent for i=1,2,....,n-1, and the vertices (x₁,y₁) and (x_n,y_n) are also adjacent.

Example:

Input: vertices[] = {{1, 1}, {4, 2}, {3, 5}, {1, 4}}
Output: 16

Input: vertices[] = {{1, 3}, {5, 6}, {2, 5}, {1, 4}}
Output: 6

Approach:

The idea is to use Shoelace formula calculates the area of a polygon as half of the absolute difference between the sum of products of x-coordinates and y-coordinates of consecutive vertices.

Shoelace formula Formula:

• The formula involves summing the products of the x-coordinates of adjacent vertices and the y-coordinates of the next adjacent vertices, and then subtracting the products of the y-coordinates of adjacent vertices and the x-coordinates of the next adjacent vertices.
• The result is multiplied by 0.5 to get the area of the polygon.

Steps-by-step approach:

• Calculate the area using the shoelace formula:
  • Initialize a variable area to 0.
  • Iterate over the vertices of the polygon.
  • For each vertex i, calculate the product of the x-coordinate of vertex i and the y-coordinate of vertex i+1, and subtract the product of the y-coordinate of vertex i and the x-coordinate of vertex i+1.
  • Add the result to the area variable.
• Take the absolute value of the area:
  • The shoelace formula can result in a negative area if the polygon is oriented clockwise.
  • To ensure that the area is always positive, take the absolute value of the area variable.
• Return the absolute value of the area variable.

Below is the implementation of the above approach:

6

Time complexity: O(N), where N is the number of points.
Auxiliary Space: O(1)