Area of a circle inscribed in a regular hexagon?

In C programming, calculating the area of a circle inscribed in a regular hexagon involves understanding the geometric relationship between the hexagon and its inscribed circle. A circle inscribed in a regular hexagon touches all six sides of the hexagon at their midpoints.

Syntax

area = (3 * ? * side * side) / 4

Mathematical Formula

For a regular hexagon with side length a, the radius of the inscribed circle is r = a(?3)/2. Using the circle area formula A = ?r², we get −

The final formula becomes: Area = 3?a²/4

Example

Here's a complete C program to calculate the area of a circle inscribed in a regular hexagon −

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

int main() {
    float side = 14.0;
    float pi = 3.14159;
    float area = (3 * pi * side * side) / 4;
    
    printf("Side of hexagon: %.1f<br>", side);
    printf("Area of inscribed circle: %.2f<br>", area);
    
    return 0;
}

Side of hexagon: 14.0
Area of inscribed circle: 461.81

Example with User Input

This example demonstrates calculating the area with different hexagon side lengths −

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

int main() {
    float sides[] = {4.0, 6.0, 8.0};
    float pi = 3.14159;
    int i;
    
    printf("Area of inscribed circles:<br>");
    printf("Side\tArea<br>");
    printf("----\t----<br>");
    
    for (i = 0; i < 3; i++) {
        float area = (3 * pi * sides[i] * sides[i]) / 4;
        printf("%.1f\t%.2f<br>", sides[i], area);
    }
    
    return 0;
}

Area of inscribed circles:
Side	Area
----	----
4.0	37.70
6.0	84.82
8.0	150.80

Key Points

• The inscribed circle's radius is r = a?3/2 where a is the hexagon side length.
• The area formula 3?a²/4 is derived from the standard circle area formula ?r².
• Use math.h library for more precise calculations with M_PI constant.

Conclusion

Calculating the area of a circle inscribed in a regular hexagon is straightforward using the formula 3?a²/4. This geometric relationship is useful in various engineering and mathematical applications.