The Area of Triangles using Hero's Formula

Date: 12/13/95 at 16:1:49
From: Anonymous
Subject: Area of Triangles, octagons

I have been surfing for about a week and this is the first site I 
have visited that actually had an area like this.  I am currently 
producing (trying) a software package for facilities Management.

Question.
If a person gave three dimensions of a triangle (in feet) and at least
noted the Base dimension, without knowing the angles because the other
two line would have to intersect someplace, is there a formula that
could calculate the area?  Dimension of a roof area for example up to 
five measurements, Squares, rectangles and odd size rectangles ok, but I 
am having a problem with the other two. When you ask for the height of a 
triangle you are met with a "what?". Octagons knowing only the five 
dimensions ?

Thank you for the opportunity to at least submit this. Just trying
to present the problem in written form helps sometimes.

From.. Keswick, Ontario (40 miles north of Toronto)

Date: 12/13/95 at 22:36:55
From: Doctor Ken
Subject: Re: Area of Triangles, octagons

Hello!

Here's one of my favorite formulas in Math: it's Hero's formula 
(sometimes called Heron's formula) for the area of a triangle.  
If the 3 side lengths are a, b, and c, then let s = (a+b+c)/2.  
The s stands for semiperimeter.  Then the area of the triangle is

                     Sqrt{s(s-a)(s-b)(s-c)}.

As far as your octagon question, I'm not sure I understand what you 
mean.  Are you given only the eight side lengths?  If so, then you can't 
figure out what the area is from that information alone (an octagon 
isn't a rigid figure in the same way that a triangle is).  Perhaps you 
could write back and clarify, and then we might be able to help.

-Doctor Ken,  The Geometry Forum

Date: 12/14/95 at 8:43:4
From: Al Barber
Subject: Re: Area of Triangles, octagons

Thank you for your quick reply. You don't know how much I appreciate it.
As for the sides of an octagon, I was not thinking clearly; I meant a
five-sided figure (not even sure of what you call it).  Maybe I will 
explain a little. In calculating floor areas, architects now use so 
many different shapes that current programs only accepting squares,
rectangles, etc. are not very efficient. I have been attempting to 
calculate the square feet without the end user having to do it manually. 
I decided at this stage to limit it to 5 sides. The users of the 
software many times are lucky if they understand any math so the easier 
for them the better. My pet theory in a lot of software, was it was 
written to make the programmer's life easier, not the users'.

So if I see three dimensions it will be triangular. If I see five (and 
that is the limit for now) I might have to ?? (I really don't know). 
And then there are circular areas, so as the song says and on and on 
and on.

Regards......
Gordon Barber

Date: 12/16/95 at 13:53:56
From: Doctor Ken
Subject: Re: Area of Triangles, octagons

Hello!

>As for the sides of an octagon, I was not thinking clearly; I meant a
>five-sided figure (not even sure of what you call it).

A five-sided figure is called a pentagon, just like the building in 
Washington, D.C.

In order to calculate the area of a pentagon, you need to know more than
just the lengths of the five sides.  You also need to know the angles, 
or you need to know what the coordinates of the five points (vertices) 
of the pentagon are.  If this is a computer program, odds are pretty 
good that you can get the coordinates.  Then I think the easiest way to 
figure out the area would be to divide it up into 3 triangles and find 
their areas, and then add them together.  To find the areas of the 
triangles, you can use Hero's formula again.

Good luck!

-Doctor Ken,  The Geometry Forum