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General Area Formula

```
Date: 02/14/2002 at 22:46:39
From: Will Voorhees
Subject: General Area Formula

I heard that there was an all-inclusive formula for the area of a
square, rectangle, parallelogram, trapezoid, and triangle.  Is this
true?  I have looked everywhere, but I can't find anything.
```

```
Date: 02/15/2002 at 11:01:29
From: Doctor Ian
Subject: Re: General Area Formula

Hi Will,

Let's start with the most complicated (i.e., least symmetric) shape,
which is a trapezoid:

a
________
/        \         area = height * (a+b)/2
/__________\
b

In the case where a and b are equal, we have a parallelogram:

a
___________
/          /       area = height * (a+b)/2
/__________/
b

And the formula still works.  Note that when a = b,

(a+b)/2 = (b+b)/2

= 2*b/2

= b

If we make all the angles square, we have a rectangle:

a
_________
|         |        area = height * (a+b)/2
|_________|
b

And the formula still works.  If we make the width the same as the
height, we have a square:

a (= h)
___
|   |              area = height * (a+b)/2
|___|
b

And the formula still works! The principal difference is that when you
have a rectangle or a square, the height is trivial to find; while
when you have a trapezoid or a parallelogram, the process can be
somewhat more involved.

So, what about a triangle?  Well, if we draw the triangle so that the
base is horizontal,

a           a
/\          /|             a
/  \        / |          . /
/    \      /  |       .   /
/______\    /___|     _____/
b         b         b

then the value of the 'top base', a, is zero, so the formula gives us

area = height * (a+b)/2

= height * (0+b)/2

= (1/2) height * b

So it works for a triangle, too - if you're willing to define a
triangle as a quadrilateral with one zero-length side.

I hope this helps.  Thanks for an interesting question!

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 02/15/2002 at 22:46:24
From: Will Voorhees
Subject: General Area Formula

Thanks for the reply!  This really helped me a lot.

William Voorhees
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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