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