Just how big should a top bar hive be?

But from the bees’ point a view, an important consideration is the
VOLUME of the hive.  Just how big is the space they are planning to live
in?  Being able to calculate the volume tells you more about the size
of the cavity WITHIN the hive – and since bees are “cavity nesters” –
meaning that they will go inside this cavity and create the structure of
their nest – the honeycomb – then VOLUME is probably what the bees are
concerned with.  It doesn’t really matter how tall a tree is, or even
how big around it is – if the volume of the hollow cavity inside the
tree isn’t large enough for the bees purposes then it isn’t a suitable
home.

Now some of us can measure the time since we sat in a geometry class
in units called decades, so conjuring up a formula to calculate area or
volume might be a bit of a challenge.

Combined with the fact that most top bar hives have sloped sides in
an attempt to ask the bees not to build their honeycombs attached to the
sides of the box – which creates a trapezoid instead of a rectangle —
and now things are really complicated!

Just how do you do the math on a three-dimensional trapezoid?

Well, here’s how it works:

The formula for the area of the trapezoid is this:

A = (( a + b) / 2) x  h  -> where h is the height, and a and b are the lengths of the parallel sides, in other words, the top and bottom.

When you have that number, the area, you multiply it by the length of
the inside of the hive, and voila – now you’ve got the volume of your
top bar hive.

So if we were to calculate the area of a Gold Star Top Bar Hive – it
would look like this.  We are showing here the dimensions of our
“follower board”, since that exactly matches the interior size and shape
of the hive body itself.

a =  15 inches
b =   6.1875  inches
h =  9.5 inches

Note:  The decimal equivalent of 3/16 = .1875.  This makes the math
easier.  (For some of us, fractions were a long time ago too!)
?

So, a + b would be 15 + 6.1875 = 21.1875 inches.
Then you would divide 21.1875 by 2.  This will get you 10.59375
Multiply that by 9.5 and get 100.64.  This figure is square inches – the area of the trapezoid, i.e. the follower board.

Then, to get the volume, you multiply the area 100.64, by the length
of the interior of the hive.  A Gold Star hive is 44.5 inches long
inside, and voila` – now you know that the volume of the cavity inside a
Gold Star hive is 4478.5 cu inches.

Now we think that’s a lot of math, and apparently somebody else did
too, because some kind and smart person made it much easier – by
creating an on-line calculator that will do the math for you – all you
have to know is the dimensions of your trapezoid.

Much easier!

http://www.onlineconversion.com/object_volume_trapezoid.htm
Here is the actual formula: 

volume = L * (b1 + (b2 – b1) * h1 / h + b1) * h1 / 2
where:  Base1 (b1)
Base2 (b2)
Total Height (h)
Partial Height (h1)
Length (L)
Volume
This calculator works to calculate a “partially filled tank”, so for a
“filled tank”, you would set the partial height (h1) and total height
(h) as the same number.

So gather up your dimensions, plug them into the Online Conversion
calculator and then you’ll know a bit about what the bees are looking at
when they look at your hive!