Can you show me how Gillings derived the formula. Was it as follows?

Using your notation and assuming the formula for the volume of a pyramid:

Let VCP = Volume of complete pyramid = a x a x 1/3 x k = 4 x 4 x 1/ 3 x 12 = 64

Let VP = Volume of small virtual pyramid projected by pyramid = b x b x 1/3 x L = 2 x 2 x 1/3 x 6 = 8

Let VTP = Volume of Truncated Pyramid = VCP - VP = 64 - 8 = 56

k - L = h (or 12 - 6 = h so h = 6) where h = height of truncated pyramid

VTP = 1/3 x h [12/h x (4 x 4) - 2 x 2] = 1/3 x 6 x [ 12/6 x (4 x 4) - (2 x 2)]

VTP = 1/3 x 6 x [2 x (4 x 4) - (2 x 2)]

VTP = 2 x [(4 x 4) + (4 x 2) + (4 x 2) - (2 x 2)]

VTP = 2 x [(4 x 4) + (4 x 2) + (2 x 2)]

VTP = 2 x [16 + 8 + 4] = 56

Mark