Polygons

For a regular pentagon find the interior angle. In the figure a regular pentagon with an interior angle marked in red is shown.

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, the blue angle is the interior angle of the polygon. Suppose we consider a regular polygon with the central angle of ${40}^{\circ }$, then find the measure of the interior angle of this polygon.

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.