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### Cross-Section of a Prism

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Date: 08/02/2001 at 10:34:37
From: Shaun
Subject: How to prove the formula of a prism?

Dear Dr Maths,

I know that the formula for a prism is area of cross-section x height.
Can you please define cross-section, and also why is this the formula?
Thanks.

Sincerely,
Shaun
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Date: 08/02/2001 at 13:03:10
From: Doctor Peterson
Subject: Re: How to prove the formula of a prism?

Hi, Shaun.

Suppose your prism is sitting on a table with its base horizontal.
Then a cross section will be the shape you get by cutting it
horizontally at any height; all cross sections will be congruent to
the base (and the top as well). This property is what defines a prism.

Now suppose you want to fill the prism with unit cubes to find its
volume. You should be able to convince yourself that you can make one
layer across the bottom (that is, fill the bottom of the prism to a
depth of one unit) with a number of cubes equal to the area of the
base. (If the base is not rectangular, just cut cubes up and rearrange
them to fit the base, and the number of them will be the same as the
number of unit squares it takes to cover the base itself.)

Now repeat this as many times as necessary to fill the whole prism;
the number of layers needed will be the height of the prism. So the
height times the area of the base is the volume of the prism.

At the bottom of our FAQ on prism formulas,

http://mathforum.org/dr.math/faq/formulas/faq.prism.html

there is a link to a page on which I explained the origin of the
formulas, at least for a rectangular prism:

http://mathforum.org/dr.math/problems/brittani5.27.98.html

If you have more questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Definitions
High School Geometry
High School Higher-Dimensional Geometry

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