How to find the surface area of a pyramid which has a regular hexagonal base of edge 6 cm and a height of 8 cm?
The area of the base should be simple, so the problem is the six triangular sides. Each triangle has area , where b = 6 cm is the length of a side of the base, and l is the height of the triangle. Consider the right triangle formed by a line from the center of the base to the center of one of the sides of the base, by the center axis of the pyramid, and the height l of the side in question. Since b=6 cm, the distance from center of base to the side of the base is . Thus by the pythagorean theorem, , and . From here, you can find the area of one of the triangles. Then just multiply by 6 to get the entire lateral area, and add the area of the base for the entire surface area.
You can divide it up into 6 equilateral triangles by drawing diagonals from opposite points.
The area of each of the equilateral triangles is
So the Total Base Area is
Now for the slanting plane areas (img2):
From before, we learnt that the length of is
Also, , as given.
Therefore, can be found by Pythagoras:
is the height of the triangular plane, so we can now work out its area:
Multiplying by 6 gives us the area of all of them =
So the total surface area is