Check the picture below.
a)
so if we look at the Gazebo from the top, it looks like the regular octagon in the picture.
now, a circle has a total of 360°, so if we split it with an octagon radii, each corner of the octagon inscribed in the circle will make a triangle, so we'll end up with 8 triangles.
now, if we further split one of those triangles, like the one there in the picture, into two triangles, the circle will be split into 16 triangles, all of equal dimensions.
we know the total area of the octagon is 310.4, so one of those 16 triangles will have an area of 310.4/16, as you see in the picture.
[tex]\bf \stackrel{\textit{area of the small triangle}}{A=\cfrac{310.4}{16}\implies A=19.4}~\hspace{7em}\stackrel{\textit{area of a triangle}}{A=\cfrac{1}{2}bh}\implies 19.4=\cfrac{1}{2}(4)h \\\\\\ 19.4=2h\implies \cfrac{19.4}{2}=h\implies \boxed{\stackrel{apothem}{9.7=h}}[/tex]
b)
so, one side is 8 units long, thus all sides, namely the perimeter is 8+8+8+8+8+8+8+8, or 8*8 = 64.
[tex]\bf \textit{area of a regular polygon}\\\\ A=\cfrac{1}{2}ap~~ \begin{cases} a=apothem\\ p=perimeter\\[-0.5em] \hrulefill\\ p=64\\ A=310.4 \end{cases}\implies 310.4=\cfrac{1}{2}a(64)\implies 310.4=32a \\\\\\ \cfrac{310.4}{32}=a\implies \boxed{9.7=a}[/tex]
