Expert: Josh - 7/5/2005

Question
I truly admire my daughter's imagination, but I'm no help with her on this puzzle she created for herself (she's eight years-old):

With a height of 100 meters and a base of 35000 square meters, what is the internal square meterage of this structure from bottom to top?  

I asked her why meters instead of feet and she said meters were easier.  Easier?  Geez, she must have her mother's brilliance.  So, yes, I would like to know the answer, but also, what formula is even used to make such a calculation?  

Answer
Hi Joseph,

I see there's a little bit of confusion. Let me say this to get it out of the way. "Square meters" is always a measure of the area. I think you are referring to the dimension of the base of a triangle, so it's better to call measure this in "meters".

It's easier to understand this by drawing a diagram. Let's form a triangle by joining the points A to B, B to C, then, from C to A.
A
|
|
|
B----------------C

Let h or |AB| represent the height.
Let b or |BC| represent the length of the base.
You can always substitute values for "h" and "b" later. But, it's easier to consider the problem in general terms.

To find the area bounded by the triangle (which i presume is what you mean when you say "square meterage" of the structure, otherwise, it makes little sense), you have to multiply the height "h" by the length of the base "b", then divide it by 2.

This works only if side AB and BC are at right angle to each other (angle equals 90 degrees). Then, the area is given by the expression A=h*b/2.

Putting h=100, b=35000, A=100*35000/2=1750000 square meters...which is huge, considering that the base of the triangle is 35 km wide.

To find the perimeter (i.e., the total length of the triangular boundary), you have to add up these lengths, P=|AB|+|BC|+|CA|.

Using "Pythagoras theorem", the longest side of the triangle (namely, the "hypotenuse") is given by |CA|=square_root_of(|AB|*|AB|+|BC|*|BC|).
Let's use smaller numbers in the following example.
Suppose that |AB|=100 and |BC|=35 (instead of 35,000),
|CA|=square_root_of(100*100+35*35)=sqrt(10000+1225)=105.948 meter approximately

So, the perimeter P=100+35+105.948 or approx. 240.948 meters. Give her credit for being imaginative:)