Biggest Reuleaux Triangle inscribed within a Square inscribed in an equilateral triangle?

Here we will see the area of biggest Reuleaux triangle inscribed within a square which is inscribed in an equilateral triangle. A Reuleaux triangle is a shape formed by the intersection of three circles of equal radius, creating a curved triangle with constant width.

Syntax

float areaReuleaux(float triangleSide);

Mathematical Relationship

For an equilateral triangle with side a, the inscribed square has side length −

x = (2 - ?3) × a = 0.464 × a

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

Area = (? - ?3) × h² / 2

Example

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

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

float areaReuleaux(float a) {
    if (a <= 0) {
        printf("Error: Side length must be positive<br>");
        return -1;
    }
    
    float x = 0.464 * a; /* Side of inscribed square */
    float area = ((M_PI - sqrt(3)) * x * x) / 2;
    return area;
}

int main() {
    float triangleSide = 5.0;
    
    printf("Equilateral triangle side: %.2f<br>", triangleSide);
    printf("Inscribed square side: %.3f<br>", 0.464 * triangleSide);
    printf("Area of Reuleaux Triangle: %.5f<br>", areaReuleaux(triangleSide));
    
    return 0;
}

Equilateral triangle side: 5.00
Inscribed square side: 2.320
Area of Reuleaux Triangle: 3.79311

Key Points

• The relationship x = 0.464a comes from geometric properties of inscribed shapes.
• Reuleaux triangles have constant width, making them useful in engineering applications.
• The area formula uses ? - ?3 ? 1.414 as a key constant.

Conclusion

The area of the biggest Reuleaux triangle inscribed in a square within an equilateral triangle depends on the triangle's side length through the relationship Area = (? - ?3) × (0.464a)² / 2. This demonstrates the elegant mathematical connections between different geometric shapes.