Exercise 3 Page 625

We begin by remembering that the volume of a right prism or a right cylinder equals the area of the base multiplied by the height.

Also, notice that a prism or a cylinder are right when the lateral edges are perpendicular to the bases. Next, let's try to find the area of a prism or a cylinder that are not right solids.

Volume of a Prism

Let's consider two prisms with the same base and height such that one of them is not a right prism.

We can find the volume of both prisms by cutting them into slices, finding the area of each slice, and then adding them all. Since both prisms have the same height, both can be divided into the same number of slices.

Notice that the slices made at the same level (height) of both prisms are equal and they have the same area. In fact, they have the same area as the base. This implies that both prisms have the same volume. In consequence, the volume of the oblique prism is also $V=Bh.$

Volume of a Cylinder

Let's consider two cylinders with the same radius and height, but one of them is an oblique cylinder.

As before, we can find the volume of both prisms by cutting them into slices, finding the area of each slice, and adding them all. Again, since both cylinders have the same height, both can be divided into the same number of slices.

Once more, we see that the slices made at the same level in both cylinders are equal and so they have the same area. Even more, they have the same area as the base. Thus, both cylinders have the same volume, which implies that the volume of the oblique cylinder is $V=Bh.$

Conclusion

According to the two examples made before, we can conclude that to find the volume of a prism or cylinder that is not a right prism or right cylinder, we multiply the area of the base by the height, just like it was a right solid.