To determine the relationship between the area of a regular hexagon and a square when both shapes have sides of length a, we first need to calculate the area of each shape.
The area of a square with side length a is straightforward. It is calculated using the formula:
Area of Square = a²
Now, let’s move on to the regular hexagon. A regular hexagon can be thought of as being composed of six equilateral triangles. The area of one equilateral triangle with side length a is given by:
Area of One Triangle = (√3/4) * a²
Since a hexagon consists of six of these triangles, the total area of the regular hexagon is:
Area of Hexagon = 6 * (√3/4) * a² = (3√3/2) * a²
Now, to compare the two areas:
The area of the square is a² and the area of the hexagon is (3√3/2) * a². To express the area of the hexagon in relation to the area of the square, we can compute:
Area of Hexagon / Area of Square = (3√3/2) * a² / a² = 3√3/2
This shows that the area of the regular hexagon is approximately 2.598 times the area of the square when both have sides of length a.