Polygons

A rectangle-shaped land has dimensions $3\times 5\mathrm{cm}$ on a map with scale $1:2000$. The owner increased the size of his land by buying some land from his neighbor. The new land has dimensions $4\times 5\mathrm{cm}$ on the map. Find the actual increase of the perimeter of the land (i.e. find the increase in the length of the fence required to enclose the whole land). Give your answer in meters.

Consider a regular polygon with the central angle of ${24}^{\circ }$. In the figure the cut of a regular polygon with unspecified number of vertices is shown. The red angle is the central angle of the polygon. Find the number of diagonals in this polygon.

The parallelogram has sides of the length $5\mathrm{cm}$ and $4\mathrm{cm}$ (see the picture). The area of this parallelogram is $S=10\sqrt{2}{\mathrm{cm}}^2$. Find the measure of the smaller of the interior angles.

Consider the rectangle $ABCD$ with length and height $|AB|=6\mathrm{cm}$ and $|BC|=2\sqrt{3}\mathrm{cm}$, respectively. Find the measure of the angle $CAB$.