A cube is divided into six identical square pyramids, each with a perpendicular height (s) that is half the length of the base edge (2s). Show that the volume of each pyramid is one third the volume of a square prism with the same base edge and perpendicular height.

pyramid

... v = 1/3 (base area)(height)
... v = 1/3 * (2s)^2 * s = 4/3 s^3

prism
... v = l w h = 2s * 2s * s = 4 s^3

To show that the volume of each pyramid is one third the volume of a square prism with the same base edge and perpendicular height, we'll first find the volume of the pyramid, and then compare it to the volume of the prism.

Let's start by finding the volume of the square pyramid. The volume of a pyramid is given by the formula:

V = (1/3) * base area * height

In this case, the base of the pyramid is a square, with side length 2s (since the perpendicular height is half the length of the base edge).

The area of a square is given by the formula:

A = side length^2 = (2s)^2 = 4s^2

So, the base area of the square pyramid is 4s^2.

Now, let's substitute the base area and height into the volume formula:

V = (1/3) * 4s^2 * s

Simplifying, we get:

V = (4/3) * s^3

Now, let's find the volume of the square prism. The volume of a prism is given by the formula:

V = base area * height

In this case, the base of the prism is also a square with side length 2s.

So, the base area of the square prism is 4s^2, which is the same as the base area of the square pyramid.

Now, let's substitute the base area and perpendicular height into the volume formula:

V = 4s^2 * (2s)

Simplifying, we get:

V = 8s^3

Now, let's compare the volumes of the pyramid and the prism:

Pyramid volume = (4/3) * s^3
Prism volume = 8s^3

To show that the volume of the pyramid is one third the volume of the prism, we can divide the pyramid volume by the prism volume:

(4/3) * s^3 / 8s^3

Simplifying, we get:

(4/3) * 1/8

Which further simplifies to:

4/24

And finally:

1/6

Therefore, the volume of each pyramid is indeed one third the volume of a square prism with the same base edge and perpendicular height.

I hope this explanation helps! Let me know if you have any further questions.