Area of a leaf inside a square in C Program?

To find the area of a leaf inside a square, we need to split it into parts and find the area of each part, then add the areas together. The leaf shape is formed by the intersection of two quarter circles within a square.

Syntax

area = a * a * (PI / 2 - 1);

Where a is the side length of the square and PI is approximately 3.14159265.

Mathematical Approach

To calculate the area, we split the leaf into two identical parts. For one part (AECA), we find the area of the quarter circle AECDA and subtract the area of triangle ACDA from it −

• Area of quadrant: ¼ × (? × r²) where r = a
• Area of right triangle: ½ × base × height = ½ × a²
• Formula: Area of leaf = a² × (?/2 - 1)

Example

The following program calculates the area of a leaf inside a square with side length 12.3 units −

#include <stdio.h>
#define PI 3.14159265

int main() {
    float a = 12.3;
    float area = a * a * (PI / 2 - 1);
    printf("Side length: %.1f<br>", a);
    printf("Area of leaf: %.6f square units<br>", area);
    return 0;
}

Side length: 12.3
Area of leaf: 86.355782 square units

Key Points

• The leaf area is always (?/2 - 1) times the square of the side length
• For any square with side a, the leaf occupies approximately 57.08% of the square's area
• The formula works for any positive value of the square's side length

Conclusion

The area of a leaf inside a square can be calculated using the formula a² × (?/2 - 1), where a is the side length. This geometric problem demonstrates the intersection of circular and rectangular areas in coordinate geometry.