Solution: Let the side length of a regular octagon be a, then the lengths of two right angles of a truncated angle are both (4-a)÷2, and the length of the hypotenuse of a truncated angle is a. According to Pythagorean theorem, the solution of (4-a) ÷ 2+(4-a) ÷ 2 = a 2 can be obtained.

It can be concluded that the side length of an octagon is 8a=32 (2- 1 under the root sign).

The cross-sectional area of the triangle is 4 (2-1under the root number) × 4 (2-1under the root number) ÷ 2 = 24- 16 under the root number.

The area of the four triangles cut out is: 2 under 96-64 roots.

The square area is: 16.

So the area of a regular octagon is16-(2 under 96-64 roots) = 2-80 under 64 roots.

Answer: The circumference of a regular octagon is 32 (2- 1 under the root sign) and the area is 2-80 under the root sign 64.