Area of squares formed by joining mid points repeatedly in C Program?

In this problem, we create a series of squares by repeatedly connecting the midpoints of the sides of the previous square. Given an initial square with side length 'a' and the number of iterations 'n', we need to find the area of the nth square.

Syntax

float calculateNthSquareArea(float side, int n);

Mathematical Formula

When we connect the midpoints of a square, the resulting inner square has an area that is half of the original square. This pattern continues for each iteration −

• 1st square: Area = a²
• 2nd square: Area = a²/2
• 3rd square: Area = a²/4
• nth square: Area = a²/2^(n-1)

Example

Here's a complete C program to calculate the area of the nth square −

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

float calculateNthSquareArea(float side, int n) {
    if (side < 0 || n <= 0) {
        return -1; // Invalid input
    }
    
    float area = (side * side) / pow(2, n - 1);
    return area;
}

int main() {
    float side = 20.0;
    int iterations = 10;
    
    printf("Initial square side length: %.2f<br>", side);
    printf("Number of iterations: %d<br>", iterations);
    
    float result = calculateNthSquareArea(side, iterations);
    
    if (result == -1) {
        printf("Invalid input values!<br>");
    } else {
        printf("Area of %dth square: %.5f<br>", iterations, result);
    }
    
    // Show progression for first few squares
    printf("\nProgression:<br>");
    for (int i = 1; i <= 5; i++) {
        printf("Square %d: Area = %.4f<br>", i, calculateNthSquareArea(side, i));
    }
    
    return 0;
}

Initial square side length: 20.00
Number of iterations: 10
Area of 10th square: 0.78125

Progression:
Square 1: Area = 400.0000
Square 2: Area = 200.0000
Square 3: Area = 100.0000
Square 4: Area = 50.0000
Square 5: Area = 25.0000

Key Points

• Each new square has half the area of the previous square
• The formula follows a geometric progression with ratio 1/2
• Input validation ensures positive side length and iteration count
• The pow() function from math.h calculates 2^(n-1)

Conclusion

The area of squares formed by connecting midpoints follows the pattern a²/2^(n-1), where each iteration halves the area. This geometric relationship makes the calculation straightforward using the power function.