Biggest Reuleaux Triangle within a Square which is inscribed within a Circle?

Here we will see the area of biggest Reuleaux triangle inscribed within a square, that square is inscribed inside one circle. The side of the square is a and the radius of the circle is r. As we know that the diagonal of the square is the diameter of the circle.

Syntax

float areaReuleaux(float radius);

Mathematical Relationship

Since the diagonal of the square equals the diameter of the circle −

2r = a?2
a = r?2

The height of the Reuleaux triangle equals the side of the square, so h = a = r?2. The area formula for a Reuleaux triangle is −

Area = (? - ?3) × h² / 2
     = (? - ?3) × (r?2)² / 2
     = (? - ?3) × 2r² / 2
     = (? - ?3) × r²

Example

This program calculates the area of the biggest Reuleaux triangle within a square inscribed in a circle −

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

float areaReuleaux(float r) {
    if (r < 0) 
        return -1;
    
    float area = (M_PI - sqrt(3)) * r * r;
    return area;
}

int main() {
    float radius = 6.0;
    float result = areaReuleaux(radius);
    
    if (result == -1) {
        printf("Invalid radius!<br>");
    } else {
        printf("Circle radius: %.1f<br>", radius);
        printf("Square side: %.2f<br>", radius * sqrt(2));
        printf("Area of Reuleaux Triangle: %.4f<br>", result);
    }
    
    return 0;
}

Circle radius: 6.0
Square side: 8.49
Area of Reuleaux Triangle: 50.7402

Key Points

• The Reuleaux triangle is a curve of constant width formed by three circular arcs.
• Its height equals the side of the inscribed square: h = r?2.
• The area formula simplifies to (? - ?3) × r² where r is the circle's radius.

Conclusion

The area of the biggest Reuleaux triangle within a square inscribed in a circle depends only on the circle's radius. The mathematical relationship (? - ?3) × r² provides an elegant solution for this geometric problem.