Biggest Reuleaux Triangle within A Square?

Here we will see the area of biggest Reuleaux triangle inscribed within a square. The side of the square is 'a'. And the height of the Reuleaux triangle is h.

The height of the Reuleaux triangle is same as a. So a = h. The area of Reuleaux triangle inscribed in a square is −

Syntax

Area = (? - ?3) × a² / 2

Where 'a' is the side of the square.

Example

Let's calculate the area of the biggest Reuleaux triangle that can fit inside a square −

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

float areaReuleaux(float a) {
    if (a < 0) {
        return -1; // Invalid input
    }
    // Area formula: (? - ?3) × a² / 2
    float area = ((3.14159 - sqrt(3)) * (a) * (a)) / 2;
    return area;
}

int main() {
    float side = 8;
    float area = areaReuleaux(side);
    
    printf("Side of square: %.2f<br>", side);
    printf("Area of Reuleaux Triangle: %.4f<br>", area);
    
    return 0;
}

Side of square: 8.00
Area of Reuleaux Triangle: 45.1024

Key Points

• The Reuleaux triangle is a curve of constant width, making it fit perfectly in a square.
• The height of the inscribed Reuleaux triangle equals the side of the square.
• The area formula uses ? and ?3 constants for precise calculation.

Conclusion

The area of the biggest Reuleaux triangle within a square is calculated using the formula (? - ?3) × a² / 2, where 'a' is the side length of the square. This geometric relationship provides an elegant solution for space optimization problems.