How to find the maximum number of triangles or hexagons that can fit in a given

Question

How to find the maximum number of triangles or hexagons that can fit in a given space.

The space is rectangular and has area WxL. Find a formula to find the maximum number of shapes of a given area, A, for

a) triangles arranged as:

with A=1/2*b*h

b) hexagons arranged as:

with A=(3sqrt(3)/2)a^2

I think what is most confusing me is how to calculate the half-squares and half-hexagons left over, so I can't just find the number in the W direction and the number in the L direction as with squares. What can I do to find the maximum number of each shape in terms of A, W, and L?

Explanation / Answer

Area of a triangle = A1 = 1/2 * b * h
Area of a hexagon = A2 = (3sqrt(3)/2 ) a^2

Area of rectangular space = W * L

Height of each triangle = h
Base of each triangle = b

Since the "height" to "width" of a regular hexagon is equal to 3/2.

Let's assume height of each hexagon = 3/2 * a
and Width of each hexagon = a

Suppose maximum n1 triangles or maximum n2 hexagons can fit in a the given rectangular space.

So, incase of triangles, L = n1 * h and W = n1 * b
   Now, n1 * A1 = W * L
   =>   n1 * 1/2 * b * h = n1 * b * n1 * h
   =>   n1 = 1/2
   Hence, W * L = 1/2 * A, where A is the Area of a triangle

Incase of hexagons, L = n2 * 3/2 * a and W = n2 * a
   Now, n2 * A2 = W * L
   =>   n2 * ((3sqrt(3)/2 ) a^2) = n2 * a * n2 * 3/2 * a
   =>   n2 = 3
   Hence, W * L = 3 * A, where A is the Area of a hexagon

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