The pyramid consists of an octagonal base and eight isosceles triangles.

Let the centre of the base be A. Any apothem will meet a side of the octagon at the midpoint of that side.

Let that midpoint be B. Let one end of that side be C

Then ABC is a right-angled triangle. It is one of 16 similar triangles comprising the octagon.

The angle CAB = 2pi/16 = pi/8

Pyramid apothem length is AB = 1.5 yards

So CB = 1.5 tan (pi/8)

Apply half-angle formula tan (a/2) = (1-cos a)/sin a

 tan (pi/8) = [1-cos(pi/4)]/sin (pi/4)

= [1 -(1/sqrt 2)]/(1/sqrt 2)

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