Biggest Square that can be inscribed within an Equilateral triangle?

Here we will find the side length and area of the biggest square that can be inscribed within an equilateral triangle. Given an equilateral triangle with side length 'a', we need to determine the maximum square that fits inside it.

Syntax

float squareSide = a / (1 + 2/sqrt(3));
float squareArea = squareSide * squareSide;

Where 'a' is the side of the equilateral triangle, and the formula derives from geometric relationships when the square is optimally positioned.

Mathematical Formula

For an equilateral triangle with side 'a', the side of the inscribed square 'x' is −

x = a / (1 + 2/?3)

The area of the inscribed square is −

Area = x² = a² / (1 + 2/?3)²

Example

Let's calculate the side length and area of the biggest square inscribed in an equilateral triangle −

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

float squareSide(float a) {
    if (a <= 0) {
        printf("Invalid input: side length must be positive<br>");
        return -1;
    }
    return a / (1 + 2/sqrt(3));
}

float squareArea(float side) {
    return side * side;
}

int main() {
    float a = 6.0;
    float side = squareSide(a);
    
    if (side > 0) {
        float area = squareArea(side);
        printf("Triangle side length: %.2f<br>", a);
        printf("Inscribed square side: %.4f<br>", side);
        printf("Inscribed square area: %.4f<br>", area);
    }
    
    return 0;
}

Triangle side length: 6.00
Inscribed square side: 2.7846
Inscribed square area: 7.7540

Key Points

• The square is positioned with one side along the base of the triangle for maximum area.
• The formula involves the constant 2/?3 which comes from the 30-60-90 triangle geometry.
• Input validation ensures the triangle side length is positive.

Conclusion

The biggest square inscribed in an equilateral triangle has side length a/(1 + 2/?3) where 'a' is the triangle's side. This geometric relationship provides an efficient way to calculate the maximum inscribed square area.