Since the base is a regular hexagon, each of the six triangles the base is divided into is an equilateral triangle. Each side of the base is equal to the length of the line from the center to the end of a side. And that means that the altitude shown divides the triangle into two "30- 60 right triangles". In particular, one leg of that right triangle (the one on the side of the hexagon) is half the length of the hypotenuse. If we call the side of the hexagon "s", the hypotenuse has length 2s and so , by the Pythagorean theorem, \(\displaystyle s^2+ (2\sqrt{3})^2= (2s)^2\) or \(\displaystyle 4s^2- s^2= 3s^2/4= 12\). Find "s" and you can find the area or all the triangles in the base and so the area of the base. Also, once you have s, you know the base of each of triangles in the upright portion and can use the Pythagorean theorem again to find the altitude of each triangle.