May someone please explain how to fill out the table ?

You have the formula written above the table.

[tex] \theta = \dfrac{s}{r} [/tex]

where
[tex] \theta = measure ~of ~central ~angle ~in ~radians [/tex]
s = arc length
r = radius

The third lines of both tables need the angle. Since the formula is already solved for theta, the central angle, just plug in s and r and calculate theta.

Left table, third line

[tex] \theta = \dfrac{s}{r} = \dfrac{8 ~in.}{6~in.} = \dfrac{4}{3} [/tex]

Right table, third line

[tex] \theta = \dfrac{s}{r} = \dfrac{5~in.}{8~in.} = \dfrac{5}{8} [/tex]

For the first line of both tables, you are looking for the arc length. Solve the formula for s, arc length.

[tex] \theta = \dfrac{s}{r} [/tex]

[tex] s = \theta r [/tex]

Left table, first line

You have the radius, r = 8 in., and theta, but theta is in degrees. We need theta in radians.

[tex] \theta = 270^\circ \times \dfrac{\pi ~rad}{180^\circ} [/tex]

[tex] \theta = \dfrac{3 \pi}{2} ~ rad [/tex]

[tex] s = \theta r = \dfrac{3 \pi}{2} \times 8 in. = 12 \pi ~in. [/tex]
(You're correct.)

Right table, first line

[tex] s = \theta r = \dfrac{2 \pi}{3} \times 3 ~cm = 2 \pi ~cm [/tex]

For the second line of both tables, you are solving for the radius. We now solve the formula for r.

[tex] \theta = \dfrac{s}{r} [/tex]

[tex] r \theta = s [/tex]

[tex] r = \dfrac{s}{\theta} [/tex]

Left table, second line

[tex] r = \dfrac{s}{\theta} = \dfrac{1.5 ~cm}{1.05} = \dfrac{10}{7} ~cm [/tex]

Right table, second line

[tex] r = \dfrac{s}{\theta} = \dfrac{9 ~cm}{5} = \dfrac{9}{5} ~cm = 1.8 ~cm [/tex]