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.