If each side of the octagon be a, then find the shortest diagonal of the octagon.

In the following figure the regular octagon has some shaded area and some unshaded area. Find the ratio of the shaded area to the unshaded area.

In the following figure a square inscribes $4$ congruent kites and 4 congruent isosceles triangles. The area of each kite is same as the area of each triangle. Find the ratio of the longer diagonal to the shorter diagonal of each kite

In the following figure two congruent squares are placed in such a way that the diagonals of one square are parallel to the sides of the other square and all the four diagonals intersect at the same point. If the area of the shaded region is $4{ \mathrm{cm}}^2$, find the area of the common region.

Find the angle $\mathrm{AGE}$ in the regular dodecagon shown below.