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Ken Ward's Mathematics Pages

Prisms (Volumes)

A prism is a solid made up of a base of a certain shape and a top which is parallel to the base and of the same shape and size. The sides of the prism are parallelograms. Examples of prisms include, cubes, cuboids and parallelepipeds (although they are normally called prisms, because they have their own names).

Prisms have cross-sections which are equal, and their sides are parallelograms. When all the sides are rectangles, the prism is called a uniform prism or a right-prism. A cuboid is one example of a right-prism.

A cylinder is similar to a prism where the polygonal base has become a circle.

The volume, V, of a prism is:
Where A is the cross-sectional area, or equivalently, the area of the base or of the top.

The base of a prism can be any polygon.

The diagram below on the right consists of three cross-sections, which are equal. Its volume is three times the volume of the sections.

If we push the sections to the right, as in the diagram below and on the right (when it is technically no longer a prism), the volume is the same as that of the prism on the left, because the sections remain unchanged in shape or volume.
The figure on the right, above, is not a prism because it has jagged edges, and these represent an error in a prism slanted to one side. The diagrams below show sections with less height than those above:
Clearly the sections of the figure on the right, above, is closer to that of a slanted prism (oblique). The diagram on the right takes this process to the extreme where it represents a prism where the sections are extremely small but approach infinity in number. Just as the volume remains the same in all the alterations of the prism, the volume of the slanted prism is the same as that of the original right-prism, which is Ah.

Because the prism above could be any prism, the volume of any prism is the area of the base times the height.

Ken Ward's Mathematics Pages

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