Area of a square inscribed in a circle which is inscribed in an equilateral triangle in C Program?

The program finds the area of a square inscribed in a circle which is inscribed in an equilateral triangle. When a circle is inscribed in an equilateral triangle of side length a, the radius of the circle is a/(2?3).

The diameter of the inscribed circle becomes the diagonal of the square: d = 2 * r = a/?3. Using the formula for the area of a square given its diagonal (1/2) * d², we get: Area = (1/2) * (a²/3) = a²/6.

Syntax

area = (a * a) / 6;

Example

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

#include <stdio.h>

int main() {
    float area, a = 10;
    
    /* Calculate area using formula: a²/6 */
    area = (a * a) / 6;
    
    printf("Side of equilateral triangle: %.2f<br>", a);
    printf("Area of inscribed square: %.4f<br>", area);
    
    return 0;
}

Output

Side of equilateral triangle: 10.00
Area of inscribed square: 16.6667

Mathematical Derivation

Conclusion

The area of a square inscribed in a circle which is inscribed in an equilateral triangle is simply a²/6, where a is the side length of the triangle. This elegant formula combines geometric relationships between all three shapes.