How to figure the angles of a circle using 4' timbers

The angle you will cut (measured from the end) will be:

90 - (180 - 360/N) / 2
which can be simplified to
180/N

where N is the number of sides in the regular polygon you create.

For instance, if you use 12 4' timbers, the angle will be 15 degrees.

Do you have a miter saw?

If so, cut your 4 foot boards at 15 degrees from vertical, and use 12 of them.

I got these 15 degrees by dividing the center angle (360 degrees) by 12; this gives me 12 isosceles triangles. The angles of the legs of the isosceles triangles are 75 degrees. If you have a miter saw that is set up to cut perpendiculars (90 degrees) the miter angle is then 15 degrees.

The diameter of your sandbox will be slightly less than 16 feet, but not by much (about 15'4")

BTW the shape you'll be creating is a dodecagon.

You could create a thirteen sided-gon and get your 16 foot diameter, but the cut angles will be much messier.

Rap

n=360/12=30

d=(180-30)/2=75

c=90-75=15