Area of the Largest Triangle inscribed in a Hexagon in C++

In this problem, our task is to find the area of the largest triangle that can be inscribed in a regular hexagon. Each side of the hexagon and triangle is of length a and b, respectively.

Area of the Largest Triangle Inscribed in a Hexagon

You can calculate the area of the largest triangle inscribed in a regular hexagon using the following formula:

Derivation

The first step is to establish a relation between a and b, as the value of a(side of the hexagon) is given.

In triangle EXF,
EF<sup>2</sup> = FX<sup>2</sup> + XE<sup>2</sup> (Using Pythagorean theorem)
=> a<sup>2</sup> = (a/2)<sup>2</sup> + (b/2)<sup>2</sup> (X is perpendicular bisector of AE and OF)
=> b = a * sqrt(3)

Now, we need the height of the triangle AEC.

Height of equilateral triangle = (b * sqrt(3)) / 2

Calculating the area of the triangle (AEC) inscribed inside a hexagon:

Area of triangle = (1/2) * base * height
Area = (1/2) * b * ((b * sqrt(3)) / 2)
Substituting b = a * sqrt(3):
Area = (1/2) * (a * sqrt(3)) * ((a * sqrt(3)) / 2)
<strong>Area = (3 * sqrt(3) / 4) * a^2</strong>

Let's see some input-output scenarios for a better understanding:

Scenario 1

<strong>Input:</strong> a = 6
<strong>Output:</strong> Area = 46.765
<strong>Explanation:</strong>
Using the formula: Area = (3 * sqrt(3) / 4) * a^2
Area = (3 * sqrt(3) / 4) * 6^2
Area = (3 * 1.732 / 4) * 36
Area = 46.765

Scenario 2

<strong>Input:</strong> a = 10
<strong>Output:</strong> Area = 129.903
<strong>Explanation:</strong>
Area = (3 * sqrt(3) / 4) * 10^2
Area = (3 * 1.732 / 4) * 100
Area = 129.903

C++ Program for Area of Largest Triangle in Hexagon

To calculate the area of the Largest Triangle inscribed in a Hexagon using a C++ program, we just need to implement the above-discussed formula as shown below:

#include <iostream>
#include <cmath>
using namespace std;

float maxTriangleArea(float a) {
  if (a < 0) // if value is negative it is invalid
      return -1;
   float area = (3 * sqrt(3) * pow(a, 2)) / 4;
   return area;
}
int main() {
   float a = 6;
   cout << "Side of hexagon is: " << a << endl;
   cout << "Area of largest triangle is: " << maxTriangleArea(a);
}

The output of the above code is as follows:

Side of hexagon is: 6
Area of largest triangle is: 46.7654