A square pyramid and a rectangular pyramid have the same height. The base area of the square pyramid is half the base are of the rectangular pyramid. Compare their volumes.

I think that the volume of the square pyramid is 1/2 the volume of the rectangular pyramid because 2 squares make a rectangle. Am I right? Thank you!

Your answer is correct, but your reasoning isn't exactly correct.

The volume of the square pyramid is 1/2 the volume of the rectangular pyramid because the base area of the square is 1/2 of the base area of the rectangle.

As per the formula V = (1/3)*h*A, you can obtain the above answer.

However, this does not necessarily mean that '2 squares of the base will make the rectangle'.

Let me give an example,

Imagine the square base has side 2 (and hence area 4), and the rectangle has lengths 1 and 8 (hence area 8). The area of the square is 1/2 times that of the rectangle but two of the squares in this case do not make the rectangle together.

good explanation.

No, you are not correct. The volume of the square pyramid is not necessarily half the volume of the rectangular pyramid just because two squares make a rectangle.

To compare the volumes of the two pyramids, we need to use the formulas for the volume of a pyramid.

The volume of a pyramid is given by the formula:
Volume = (1/3) * Base Area * Height

Let's call the base area of the square pyramid A and the base area of the rectangular pyramid B. Since the base area of the square pyramid is half the base area of the rectangular pyramid, we can express this as A = (1/2) * B.

Both pyramids have the same height, so we can call the height of both pyramids h.

Now, let's compare the volumes of the two pyramids.

Volume of square pyramid = (1/3) * A * h
Volume of rectangular pyramid = (1/3) * B * h

Substituting A = (1/2) * B, we have:
Volume of square pyramid = (1/3) * ((1/2) * B) * h
= (1/6) * B * h

The volume of the square pyramid is (1/6) times the volume of the rectangular pyramid, not half.

To compare the volumes of the square pyramid and the rectangular pyramid, let's establish some formulas and calculations.

First, let's denote the height of both pyramids as 'h', and the base area of the square pyramid as 'A_s' and the base area of the rectangular pyramid as 'A_r'.

The volume of a pyramid is given by the formula:
Volume = (1/3) * base area * height

For the square pyramid, the volume can be calculated as:
Volume_s = (1/3) * A_s * h

For the rectangular pyramid, the volume is:
Volume_r = (1/3) * A_r * h

You stated that the base area of the square pyramid is half the base area of the rectangular pyramid. So we have:
A_s = (1/2) * A_r

Substituting this into the volume formula for the square pyramid, we get:
Volume_s = (1/3) * (1/2) * A_r * h
= (1/6) * A_r * h

Comparing the volumes of the two pyramids, we can write:
Volume_s / Volume_r = [(1/6) * A_r * h] / [(1/3) * A_r * h]
= (1/6) / (1/3) (canceling out common terms)
= 1/2

Therefore, the volume of the square pyramid is half the volume of the rectangular pyramid. So, your conclusion is correct.

Feel free to ask if you have any further questions!