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### Parallelepipeds

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Date: 05/06/97 at 01:27:54
From: Amanda
Subject: Trapezoid formula

What is the formula for finding the lateral area and the total surface
area of a trapezoid?
```

```
Date: 05/06/97 at 08:03:57
From: Doctor Jerry
Subject: Re: Trapezoid formula

Hi Amanda,

I'm not certain what you mean by a trapezoid.  A trapezoid is a
quadrilateral with certain properties, and it is a plane figure.
This means that it is not a three-dimensional figure, so it can't have
lateral and total surface areas.  Perhaps you meant to ask about a
parallelepiped, which is a three-dimensional figure?

A parallelpiped is often defined by choosing a vertex and giving three
vectors; the lengths and directions of these vectors determine the
three edges of the parallelepiped meeting at that vertex. There are
formulas, usually expressed in terms of vectors, for determining the
total surface area and lateral area (I'm not sure what you mean by
this).

I'll give an example. Suppose a parallelepiped is resting on the
(x,y)-plane, with one vertex at the origin. We may assume that one
side lies along the y-axis. I'll use the vector a = (0,p,0) (or you
can think about it as the point with coordinates (0,p,0)) for this
side. The second side will be b = (q,r,0). In my visualization I'm
thinking about q < 0. Now complete the parallelogram in the (x,y)-
plane, with a and b as adjacent sides.  The third edge from (0,0,0)
will be c = (s,t,u). The numbers s and u are negative. The three
vectors/points a, b, and c define the parallelepiped.

The area of the base is just the length of a times the length of b
times the sine of the angle between a and b. This is the length of
the "cross product" of the vectors a and b, written as |a x b|. I'll
use this notation in the remainder of my comments. If you don't know
vectors, then just think of |a x b| as the length of a times the
length of b times the sine of the angle between a and b.

Okay, the lateral surface area of the parallelepiped is:

2(|a x c| + |b x c|)

The total surface area is this plus the areas of the top and bottom.
So, the total surface area is:

2(|a x c| + |b x c|) + 2|a x b|

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Linear Algebra

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