Biggest Reuleaux Triangle inscribed within a square which is inscribed within an ellipse?

Here we will see how to calculate the area of the biggest Reuleaux triangle inscribed within a square, where that square is inscribed inside an ellipse. We know that the major axis length is 2a, and the minor axis length is 2b. The side of the square is 'x', and the height of the Reuleaux triangle is h.

Mathematical Formula

We know that the side of a square inscribed in an ellipse with major axis 2a and minor axis 2b is −

x = 2ab / ?(a² + b²)

The height of the Reuleaux triangle is same as the side of the square, so h = x. The area of a Reuleaux triangle with height h is −

Area = (? - ?3) × h² / 2

Syntax

float areaReuleaux(float a, float b);

Example

This program calculates the area of the biggest Reuleaux triangle inscribed within a square that is inscribed within an ellipse −

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

float areaReuleaux(float a, float b) {
    if (a <= 0 || b <= 0) {
        return -1;  /* Invalid input */
    }
    
    /* Side of square inscribed in ellipse */
    float x = (2 * a * b) / sqrt((a * a) + (b * b));
    
    /* Area of Reuleaux triangle with height = x */
    float area = ((3.14159 - sqrt(3)) * x * x) / 2;
    
    return area;
}

int main() {
    float a = 5;  /* Semi-major axis */
    float b = 4;  /* Semi-minor axis */
    
    printf("Semi-major axis (a): %.1f<br>", a);
    printf("Semi-minor axis (b): %.1f<br>", b);
    printf("Area of Reuleaux Triangle: %.6f<br>", areaReuleaux(a, b));
    
    return 0;
}

Semi-major axis (a): 5.0
Semi-minor axis (b): 4.0
Area of Reuleaux Triangle: 10.179104

Key Points

• The formula uses the relationship between ellipse dimensions and the inscribed square.
• A Reuleaux triangle is a curve of constant width formed by the intersection of three circles.
• The calculation involves both the geometric properties of ellipses and the special characteristics of Reuleaux triangles.

Conclusion

The area of the biggest Reuleaux triangle inscribed in a square within an ellipse depends on both semi-axes of the ellipse. The formula combines elliptical geometry with the unique properties of Reuleaux triangles to provide the maximum achievable area.