SOLUTION: The base of a right pyramid is a regular hexagon with sides of length 10 m. The altitude is 5 m. Find the total surface area of the pyramid.

0.  Make a sketch to follow my arguments.

    Draw the pyramid, its base and its altitude.


1.  The total surface area is the sum of the base area and the lateral area.


    The base area is the area of the regular hexagon with the side length of 10 m.

    This hexagon is the union of 6 regular triangles with the side length of 10 m.

    Hence, the base area is  =  .


2.  The lateral area is 6 times the area of one lateral face. 
    
    Each lateral face is a triangle with the base of 10 m long.

    Consider the slant height of the pyramid, which is the height of the triangular lateral face.

    This slant height is the hypotenuse of the right-angled triangle with the legs of 5 m (the altitude of the pyramid) and 

     =  (the apothem of the regular hexagon at the base).

    Hence, the slant height length is  = 10 m.

    Then the area of each triangular lateral face is  = 50 m^2 and the area of 6 such lateral faces is 6*50 = 300 m^2.


4.  Thus the total surface area of the pyramid is   .