each of the isosolese triangles will look like the diagram. all we need to do is find the base of the triangle, and then multiply it by N to find the approximation of the arc length, since it is N of these bases will be used to approximate it.
Let the base of each triangle be BC as indicated by the diagram.
since the angle that subtends the arc is 3pi/4, the angle at the vertex of each triangle will be 3pi/4N, since we would divide the angle into N equal peices. if we draw a vertical line that bisects the base, it will also bisect the angle at the vertex since the triangle is isoseles. now we can find the length of the base using the sine ratio, since we obtain two right-triangles when we bisect the isosolese, each with an angle of 1/2 * 3pi/4N = 3pi/8N at the vertex.
recall that the radius of the circle is 12, so the hypotenuse will be 12. the base will be 1/2 BC, which will be the opposite side. so we have:
sin(3pi/8N) = opp/hyp = [(1/2)BC]/12
=> sin(3pi/8N) = BC/24
=> BC = 24sin(3pi/8N)
so the length of each of the bases will be 24sin(3pi/8N), so the length of N bases will be 24Nsin(3pi/8N) as desired
tell me if you don't get my explanation