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Find the area of largest Triangle inscribed in a hexagon we need to learn these figures are and how 1 is inscribed inside other.

**Triangle** is a closed figure with 3 sides which may be equal or different size.

**Hexagon** is a closed figure with 6 sides which may be equal or unequal in size.

A triangle inscribed inside a hexagon has all its vertices touching vertices of hexagon. So, the sides of the triangle can be treated as diagonals of a *regular hexagon*. The hexagon considered here is a regular hexagon, which leads to make the largest triangle an Equilateral triangle.

Let’s derive the formula for this,

Please refer to the following image −

In triangle AGB, we apply pythagoras theorem.

(a/2)

^{2}+ (s/2)^{2}= a^{2},a = side of regular hexagons = side of equilateral triangle

a

a^{2}/4 + s^{2}/4 = a^{2}^{2}- a^{2}/4 = s^{2}/4 3a^{2}/4 = s^{2}/4 3a^{2}= s^{2}a√3 = s Area = (3√3*a^2)/4

Let’s take an example,

Side of hexagon = 6

Area of triangle = 46.7654

Explanation, Area = 3√3*36/4 = 46.7654

#include <iostream> #include <math.h> using namespace std; int main() { float a = 6; if (a < 0) cout<<"Wrong Input!"; float area = (3 * sqrt(3) * pow(a, 2)) / 4; cout <<"The area of the triangle is "<<area; return 0; }

The area of the triangle is 46.7654

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