4 over 3 pi r cubed gives you the volume of a sphere
but where does the formula come from?
first we'll draw a perfect sphere
and fill in it's volume
next we will divide the sphere into equal sized pyramids
with square bases
so let's take a look at one pyramid
and remember that its volume is equal to one over three area of the base times height
now let's quickly show that the height of all the pyramids that make up the sphere
is actually equal to the radius
so our equation becomes one over three area of the base times radius
and we will simplify this equation by replacing radius with r
and area of the base with B
we will be numbering the bases so we start with base one and our equation is
base-1 r over 3
so in order to find the volume of the sphere
we simply have to calculate the volume of the pyramids and add them together
giving us a simple formula
so lining up the pyramids we start from the beginning
and we have
volume of the sphere
is equal to
base-1 r over three plus
base-2 r over three
plus
base-3 r over three
and we will continue this process up until the last pyramid
now we don't know how many pyramids we have so we just let base-n r over
three simply represent the last pyramid
so having added up the pyramids together
we have this complete simple formula to work with
so next we will use algebra and begin by factoring out a one over three
and we will also factor out the radius
giving us
r over three times the sum of all bases
so now let's concentrate on the sum of all bases and remember these are the
bases of the pyramids that make up the sphere
and as you can see
the bases actually form the surface area of the sphere
so the sum of all bases
is equal to the surface area
which equal to four pi r squared
now a quick explanation for why the surface area is four pi r squared
is first we'll look at our sphere
and take its largest possible circumference
with that circumference will make a circle
and now the amazing fact is the surface area of the sphere
is equal to exactly four times the area of the circle
the area of each circle is pi r squared
combine them together and we get
four pi r squared
so now let's go back to our simple formula
and replace the sum of all bases with
four pi r squared
now we combined the r's together to get r cubed and
now we simply rearrange the equation to get
four over three pi r cubed
so four over three pi r cubed gives you the volume of any sized sphere
that exists