(One over three) (area of the base) times height gives you the volume of a square pyramid,
but where does the formula come from?
First we'll create a pyramid with a square base and fill in its volume.
Next we divide the height of the pyramid into equal vertical slices and we use this as a
reference to build a step pyramid.
Now a very important thing to notice about the step pyramid, is that the more slices
it has, the closer it gets to the shape of the square pyramid.
And if the step pyramid has an infinite amount of slices, then the volume of the step pyramid
is equal to the volume of the square pyramid.
So lets derive a formula for the volume of a step pyramid with an arbitrary amount of
slices.
First we will be numbering the slices from top to bottom
and we label the bottom slice with the letter n.
Next we can see that step pyramid is actually made up of rectangular prisms.
So lets take a look at one of the prisms and remember that it's volume is equal to length
x width x height. Since the base of each prism is actually a
square then the length and width are the same.
Therefore volume is equal to length x length x height
So now we have a general formula for volume of a rectangular prism,
but we must find a specific formula that applies to all the prisms that make up the step pyramid.
First, we begin by looking at the height of the pyramid and remember we divided the height
into equal vertical slices.
Mathematically this is represented as height divided by the number of slices, which we
simplify as h/n.
Therefore each slice has a height of h/n, which is actually the height of each rectangular
prism.
Next we look at the length of the pyramid, and we divide the length of the base by the
same number slices as the height.
So we have length of the base divided by the number of slices, which we simplify as L/n.
Therefore each slice has a length of L/n.
Now if we line up all the rectangular prisms at one end, we can see that the first prism
has a length of L/n.
Looking at the second prism, its length is l/n + l/n, which simplifies to 2l/n.
And the third prism has a length of 3 L/n. As we continue, the length of each prism increases
in the same pattern all the way down to the last prisms which has a length of n times
L/n. Therefore the length of each prism depends
on the prism number. So the length of each prism is equal to (the
prism #) x L/n
And now the volume of a rectangular prism becomes equal to (prism #) L/n x (prism#)L/n
x h/n and we can simplify this as (prism#)L/n squared
times hxn
So in order to find the volume of the step pyramid, we simply calculate the volume
of all the rectangular prisms and add them together.
So lining up the prisms we start from the beginning and we have volume of the step pyramid
is equal to .... one L over n squared x h/n plus
two L over n squared x h/n plus three L over n squared x h/n
and we continue this process up until the last prism which has a volume of
n L/n squared x h/n.
So having added up all the prisms together have this complete formula to work with.
Next we will use algebra and factor out h/n and we will also factor out l/n squared.
Therefore the volume of the step pyramid is equal to l/n squared x h/n x the sum of integers
squared.
So now lets concentrate on the sum of integers squared and this formula represents a series
that contains an arbitrary amount of terms.
Amazingly this series can actually be represented by a single formula. To understand why this
is true please click on the link where I provide a geometric understanding for this formula.
So now we replace the sum of integers squared, with
n(n+1)(2n+1) over 6.
Next we will simply our formula.
First L/n squared becomes L squared over n squared.
Then we multiply this with h/n to get L squared x h over n cubed.
Next we multiply out (n+1) times (2n+1) and we have
2n^2 + n + 2n + 1 and this simplifies to 2n^2 + 3n + 1.
Now n and n cubed cancel out which leaves us with n^2 at the bottom.
Next we switch 6 and n^2.
We then distribute the denominator and we cancel out like terms
And now we have volume of the step pyramid is equal to L^2h over 6 times 2+ 3/n + 1/n^2
So now let's look back at our step pyramid and we will increase the number of slices
to infinitely many slices, therefore the volume of the step pyramid becomes equal to the volume
of the square pyramid. Now in terms of our formula, when we increase
number of slices of the pyramid we are increasing the value of n in our formula.
So lets take a look at what happens to both terms that contain n as it increases to infinity.
As you can see the values get smaller and smaller, and as we approach infinity the values
get so small that they actually become equal to zero.
So 3/n + 1/n^2 becomes 0+0.
And what we're left with is L^2h/ 6 times 2 and this simplifies to L^2 h /3
Now l^2 actually represents the area of the base of the pyramid.
Therefore we replace L squared with B, which is the area of the base.
And finally we have 1/3 area of the base x heigth which gives us the volume of a square
pyramid.