Hello, maxreality!

Your sketch is not quite right.

The height is a *vertical* measurement, not along the slanted edge.

I assume they want us to derive a formula "from scratch".

We have this cross-section of the entire pyramid.

Code:

- A
: *
: /|\
: / | \
y / |y \
: / | \
: / F| \
- B *-----*-----* D
: /: a/2 | a/2 \
h / : | \
: / : | \
- *---*-----*---------*
C H G b/2 E
: - - - - b - - - - :

is the frustum of the pyramid.

. . The lower base is:

. . The upper base is:

. . Its height is

is the entire pyramid.

. . Its base is:

. . Its height is:

Its volume is: .

Draw

Note that: .

Since .[1]

The volume of the upper pyramid is: .

The volume of the frustum is:

. .

. . . .

Substitute [1]:

. .

Therefore: . .**

What happens if ? . If ?

If , we have a "box" with a square base (side ) and height

Of course, the volume is: .

If , we have an *entire* pyramid with base area and height

Of course, the volume is: .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

**

This formula can be generalized.

. .

. . . . where are the areas of the two bases.