The total surface area(\(T_s\)) is the sum of the area of the base and the sum of the area of all the triangle sides.

\(T_s = A_b + A_s\)

Where: \(A_b\) is the area of the base and \(A_s\) is the area of the sides

Calculate the area of the base:

To calculate the central angle: \(\frac{360}{8} = 45\)

To calculate the base angles: \(\frac{180-45}{2} = 67.5\)

\(A_b =\frac{1}{2} (5^2)(\frac{\sin{67.5} \sin{67.5}}{\sin{45}}) (8) \)

Since there are 8 equal sides of an octagon, multiply the area of one triangle to 8.

\(A_b =120.71 \)

Calculate the area of the sides:

Since there are 8 sides the area of the side is \(8(\frac{bs}{2})\), where \(b\) is the lenght of the base and \(s\) is the altitude of the triangle on the side.

Calculate \(s\):

\(s^2 = x^2 + 20^2\)

\(\tan{67.5} = \frac{x}{2.5}\)

\(x = 2.5\tan{67.5}\)

\(s =20.89\)

\(A_s = 8\frac{1}{2} (5) (20.89)\)

\(A_s = 417.82\)

\(A_t = 120.71 + 417.82 = 538.53 ~cm^2\)