professor Dave again let's talk about
volume we just learned how to calculate
the two-dimensional area covered by the
surface of a three-dimensional object
but sometimes we want to do more than
that sometimes we want to calculate the
three-dimensional space that is
contained within that object which we
can think of as the amount of water we
will need to fill up the object if it
were perfectly hollow just the way the
area of a square is found by running a
side length across the shape if we take
that square surface and run it across a
third dimension we can get a cube a cube
is a prism with squares for all its
faces to find the volume of a cube
instead of just squaring the side length
which gives us the area of one square
face we cube the side length to get the
volume of the cube with a side length of
2 meters this cube has a volume of 8
meters cubed or cubic meters we can
easily visualize this by saying that if
we have some 1 by 1 by 1 cubes or cubic
units it will take 8 of them to fill
this larger cube for a rectangular prism
it's just length times width times
height 2 times 3 times 5 gives 30 cubic
meters what we are really doing is
calculating the area of the base and
multiplying by the height as though
running the base up through the height
of the shape to span its volume
this works for any type of prism like a
triangular prism here we have triangular
basis so we just get the area of one
base and multiply by the height if the
triangle itself has a base and height of
two and three meters and the prism has a
height of 10 meters we get 1/2 2 times 3
times 10 or 30 cubic meters
this works for cylinders too it's always
just base times height the base of a
cylinder is a circle so we find the area
of this circle and then multiplied by
the height of the cylinder if the height
is 6 meters and the radius is 3 we get
PI R squared or 9 pi times 6 or 54 pi
cubic meters now let's move on to some
trickier shapes how do we calculate the
volume of a pyramid this will be
different from a prism because we can't
just multiply the base times the height
as the triangular faces sloped inward
instead we will use the formula one
third base times height so if we have a
square pyramid which has a base with a
length of 3 and the pyramid has a height
of 5 we get 9 for the base times 5 is 45
times 1/3 is 15 cubic meters
what if we don't know the height of the
pyramid but only the height of the
triangles luckily we know the
Pythagorean theorem so we could imagine
a triangle where one leg is the height
of the pyramid the other leg is half the
length of the base and the hypotenuse is
the height of a triangular face we plug
them into the equation and we can then
solve for the height so it's just one
extra step cones work just like pyramids
again it will be 1/3 base times height
except that here the area of the base is
given by PI R squared so we get 1/3 PI R
squared H if we have 2 for R and 9 for H
we get 4 pi times 9 is 36 times 1/3 is
12 cubic meters
lastly we need to be able to calculate
the volume of a sphere the formula for
this is 4/3 PI R cubed so with a radius
of 3 we cube it to get 27 divided by 3
to get 9 and multiplied by 4 to get 36
times pi is 36 PI cubic meters these are
most of the regular shapes for which we
have reliable equations to calculate
volume there are of course many
irregular shapes that have a more
unpredictable curvature and to calculate
the volumes of these we will need
calculus we aren't there yet we have
lots more algebra to learn first and
since we've wrapped up most of the
crucial topics in geometry we are safe
to go back to equations with variables
right after we check comprehension
thanks for watching guys subscribe to my
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as always feel free to email me
professor Dave explains at gmail.com
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