Polygon Area Calculator - Problem

Given the coordinates of a polygon's vertices in order, calculate the area of the polygon using the Shoelace formula (also known as the surveyor's formula).

The vertices are provided as a list of [x, y] coordinate pairs. The polygon can be convex or concave, and may even be self-intersecting. For self-intersecting polygons, calculate the absolute area.

The Shoelace formula states that for a polygon with vertices (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ), the area is:

Area = ½ × |∑(xᵢ × yᵢ₊₁ - xᵢ₊₁ × yᵢ)|

where the sum runs from i = 0 to n-1 and indices are taken modulo n.

Input & Output

Example 1 — Simple Square

$ Input: vertices = [[0,0], [0,2], [2,2], [2,0]]

› Output: 4.0

💡 Note: A square with side length 2 has area = 2 × 2 = 4.0

Example 2 — Triangle

$ Input: vertices = [[0,0], [3,0], [3,4]]

› Output: 6.0

💡 Note: Right triangle with base 3 and height 4: area = ½ × 3 × 4 = 6.0

Example 3 — Complex Polygon

$ Input: vertices = [[1,1], [4,1], [4,3], [2,3], [2,2], [1,2]]

› Output: 4.0

💡 Note: L-shaped polygon calculated using shoelace formula gives area 4.0

Constraints

3 ≤ vertices.length ≤ 10⁴
-10⁶ ≤ vertices[i][0], vertices[i][1] ≤ 10⁶
Vertices are given in order (clockwise or counterclockwise)
Polygon may be self-intersecting

Visualization

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Asked in

G Google 25 a Amazon 18 M Microsoft 15 Apple 12

The key insight is using the Shoelace formula which calculates polygon area in one pass through vertices. The optimal approach applies the mathematical formula directly: sum of cross products divided by 2. Time: O(n), Space: O(1)

Common Approaches

✓ Shoelace Formula (Optimal)

⏱️ Time: O(n) Space: O(1)

Apply the shoelace formula in one pass through vertices: sum of (x[i] * y[i+1] - x[i+1] * y[i]) for all edges, then take half the absolute value.

Triangle Decomposition

⏱️ Time: O(n) Space: O(1)

Divide the polygon into triangles by connecting one vertex to all non-adjacent vertices, then calculate each triangle's area using the cross product formula.

Shoelace Formula (Optimal) — Algorithm Steps

Initialize sum to zero
For each vertex, multiply x by next y and subtract next x times current y
Add result to running sum
Return half the absolute value of total sum

Visualization

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Step-by-Step Walkthrough

Cross Products

For each edge, calculate x[i] * y[j] - x[j] * y[i]

Sum Terms

Add all cross products to running sum

Final Area

Take absolute value and divide by 2

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>

double solution(int vertices[][2], int n) {
    if (n < 3) {
        return 0.0;
    }
    
    double areaSum = 0.0;
    
    // Apply shoelace formula
    for (int i = 0; i < n; i++) {
        int j = (i + 1) % n; // Next vertex (wraps around)
        areaSum += (double)vertices[i][0] * vertices[j][1];
        areaSum -= (double)vertices[j][0] * vertices[i][1];
    }
    
    return fabs(areaSum) / 2.0;
}

int main() {
    char line[10000];
    fgets(line, sizeof(line), stdin);
    line[strcspn(line, "\n")] = 0;
    
    // Simple parsing for [[x1,y1],[x2,y2],...]
    int vertices[1000][2];
    int n = 0;
    char *ptr = line + 1; // Skip first '['
    
    while (*ptr && *ptr != ']') {
        if (*ptr == '[') {
            ptr++;
            vertices[n][0] = atoi(ptr);
            while (*ptr && *ptr != ',') ptr++;
            ptr++; // Skip comma
            vertices[n][1] = atoi(ptr);
            while (*ptr && *ptr != ']') ptr++;
            ptr++; // Skip ']'
            n++;
        }
        ptr++;
    }
    
    double result = solution(vertices, n);
    printf("%f\n", result);
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(n)

Single loop through all n vertices, constant work per vertex

✓ Linear Growth

Space Complexity

O(1)

Only stores running sum and temporary variables

✓ Linear Space

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Medium Frequency

~15 min Avg. Time

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Ln 1, Col 1

Smart Actions

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Polygon Area Calculator - Problem

Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Shoelace Formula (Optimal) — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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