1. pyramid....

1.) Find the lateral area of a regular hexagonal pyramid if each edge of base is 10m and each lateral edge is 15m.

2.) Find the volume and the lateral area of a regular triangular pyramid if the altitude is 20 inches and the base of equal sides is 9 inches.

3.) Find the volume and the total area of a triangular pyramid if each lateral edge is 30 meters and the base is 10 meters on a side. It's base is an equilateral triangle.

2. Hello, orochimaru700!

Here's the first one . . .

1) Find the lateral area of a regular hexagonal pyramid
. . .if each edge of base is 10m and each lateral edge is 15m.
I assume the "lateral edge" is what most of us call the "slant height".

One of the triangular side panels looks like this:
Code:
*
/|\
/ | \
/  |  \ 15
/  h|   \
/    |    \
/     |     \
/      |      \
* - - - * - - - *
5

From Pythagorus: . $h^2 + 5^2 \:=\:15^2 \quad\Rightarrow\quad h^2 \:=\:200 \quad\Rightarrow\quad h \:=\:10\sqrt{2}$

The area of this triangle is: . $\tfrac{1}{2}bh \:=\:\tfrac{1}{2}(10)\left(10\sqrt{2}\right) \:=\:50\sqrt{2}\text6{ m}^2$

Therefore, the area of the six panels is: . $A \;=\;6\times 50\sqrt{2} \;=\;300\sqrt{2}\text{ m}^2$