Biggest Reuleaux Triangle within a Square which is inscribed within a Right angle Triangle?

Here we will see how to find the area of the biggest Reuleaux triangle inscribed within a square, where that square is inscribed inside a right angled triangle. A Reuleaux triangle is a curved triangle with constant width formed by the intersection of three circles.

The side of the square inscribed in a right angled triangle with height l and base b is given by the formula below. The height of the Reuleaux triangle equals the side of the square.

Syntax

float areaReuleaux(float l, float b);

Where:

• l − Height of the right angled triangle
• b − Base of the right angled triangle

Mathematical Formula

The side of square inscribed in right triangle: a = (l × b) / (l + b)

Area of Reuleaux triangle: Area = ((? - ?3) × a²) / 2

Example

Here's a complete C program to calculate the area of the biggest Reuleaux triangle −

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

float areaReuleaux(float l, float b) {
    if (l <= 0 || b <= 0) {
        return -1; /* Invalid input */
    }
    
    /* Side of inscribed square */
    float a = (l * b) / (l + b);
    
    /* Area of Reuleaux triangle */
    float area = ((3.1415 - sqrt(3)) * a * a) / 2;
    
    return area;
}

int main() {
    float l = 5;  /* Height of right triangle */
    float b = 12; /* Base of right triangle */
    
    float result = areaReuleaux(l, b);
    
    if (result == -1) {
        printf("Invalid input: dimensions must be positive<br>");
    } else {
        printf("Height of triangle: %.1f<br>", l);
        printf("Base of triangle: %.1f<br>", b);
        printf("Area of Reuleaux Triangle: %.5f<br>", result);
    }
    
    return 0;
}

Height of triangle: 5.0
Base of triangle: 12.0
Area of Reuleaux Triangle: 8.77858

Key Points

• The square inscribed in a right triangle has side length a = (l × b) / (l + b)
• The Reuleaux triangle's height equals the square's side length
• Always validate input parameters to avoid invalid calculations

Conclusion

This program calculates the area of the largest Reuleaux triangle that can fit inside a square inscribed in a right triangle. The key is finding the inscribed square's side length first, then applying the Reuleaux triangle area formula.