Area of a leaf inside a square?

Here we will see how to calculate the area of a leaf-shaped region inside a square ABCD, where each side of the square has length 'a'.

The leaf consists of two equal circular segments. Each segment is formed by the intersection of a circle with the square. The area of each segment is calculated using the formula for a circular segment.

Syntax

float leafArea = a * a * (?/2 - 1);

Where 'a' is the side length of the square and ? ? 3.14159.

Mathematical Formula

The area of one segment (p) is given by −

p = (a²/4) × (? - 2)

Since the leaf has two equal segments, the total area is −

Leaf Area = 2p = a² × (?/2 - 1)

Example

#include <stdio.h>
#define PI 3.14159

float leafArea(float a) {
    return (a * a * (PI/2 - 1));
}

int main() {
    float square_side = 7.0;
    printf("Square side length: %.1f
", square_side);
    printf("Leaf Area: %.4f
", leafArea(square_side));
    return 0;
}

Square side length: 7.0
Leaf Area: 27.9667

Key Points

• The leaf area formula is derived from the intersection of two circles within the square.
• The constant (?/2 - 1) ? 0.5708 represents the geometric relationship.
• The area is always less than the square's area (a²).

Conclusion

The leaf area inside a square can be calculated using the formula a² × (?/2 - 1), where 'a' is the side length. This geometric problem demonstrates the application of circular segment area calculations in practical scenarios.