What is the area of a hexagon with side length 6 cm?

Answer 1

You can solve this by thinking of the hexagon as being six equilateral triangles, much like pie slices. If the side length of the hexagon is 6cm, then each of those three equilateral triangles has a side length of 6cm. All you need to do then, is get the area of one of those triangles, and multiply it by six.

To get the area of the triangle, break it into two equilateral triangles to get it's height. That gives you a height we'll call h, a width of 3, and a hypotenuse of 6, so we can work out the height using the Pythagorean theorem:

h = (62 - 32)1/2

h = (36 - 9)1/2

h = √27

h = 3√3

Now take that height, multiply by it's base, and divide by two, which will give you the area of the triangle:

at = 3√3 × 6 / 2

at = 9√3

We already know that the hexagon is six times the area of the triangle, so now we can work out our final answer:

a = 6at

a = 6 × 9√3

a = 54√3

You can make this method generic, substituting a variable for the number 6 and simplifying the expression. Start by assembling these parts into a single expression, using "L" instead of 6:

The height of our equilateral triangle is:

(L2 - [L / 2]2)1/2

= (L2 - L2 / 4)1/2

= (3L2 / 4)1/2

= (3L2)1/2 / 2

= √3 × L / 2

The area of our equilateral triangle is half of that height multiplied by L, it's width:

L × (√3 × L / 2) / 2

= L2√3 / 4

And so the total area of our hexagon will be six times that:

A = 6(L2√3 / 4)

A = (3√3)/2 × L2