Solution by: Andrew Ross, age 17, Dr. Challoner's Grammar School

solution.

Call the side length of the square s. Use the diagram for other lengths.

The octagon has side lengths a and b.

By Pythagoras Theorem:

r = sÖ2
2

Because s = r + d so

d = s(1 - Ö2
2
).

One side of the octagon is given by:

a = s - 2d = s - 2s(1 - Ö2
2
) = s(Ö2 - 1).

By Pythagoras Theorem b² = 2 d² so b = d Ö2. Hence

b = sÖ2 (1 - Ö2
2
) = s(Ö2 - 1) = a.

The octagon has rotational symmetry of order 4 as it was constructed from a square and all the sides are the same length, so it is a regular octagon.