13.3 Volumes of Solid Shapes

as with shapes in a plane, fundamental principles determine how volumes behave when solid shapes are moved or combind...what are those principles

1) if we move a solid shape rigidly without stretching or shrinking it, then its volume does not change 2) if we combine a finite number of solid shapes without overlapping them, then the volume of the resulting shape is the sum of the individual solid shapes

ex: what is the volume of sand in a cone shaped pile that is 15 feet high and has a radius at the base of 7 feet

according to the volume formula, the volume of sand is 1/3 x 15 x pie(7)^2 ft cubed, which is about 770 cubic feet of sand

ex: what is the volume of a 4 in tall can that has a circular base of radius 1.5 inches

area of the base is piRsq = pie(1.5)^2 which is about 7.07 in sq, therefore the volume of the can is 4 x 7.07 in ^3 = 28 in cubed

the volume formulas for prisms and cylinders is valid not only for right prisms and cylinders but also for oblique ones..why?

because an oblique prism or cylinder can be sheared into a right one. during shearing, neither the height nor the area of the base nor the volume changes, so the same formula applies to the oblique prism or cylinder as to the right prism or cylinder

how do we know the formula volume for a sphere

because volume is a three dimensional attribute, it is not surprising that the formula involves pi cubed, which has an exponent of three, also because spheres are related to circles, it is not surprising that the formula involves pi

what's the height of a prism or cylinder

distance between the planes containing the two bases of the prism or cylinder, measured in the direction perpendicular to the bases

if you have a solid lump of clay, you can mold it into various different shapes, explain

each of these different shapes is made of the same volume of clay, from the point of view of the moving and additivity principles it is as if the clay had been subdivided into many tiny pieces, and then these tiny pieces were recombined in a different way to form a new shape, therefore the new shape is made of the same volume of clay as the old shape

in the shearing process,

each thin slice remains unchanged: each slice is just slid over and is not compressed or stretched

The volume Formula for prisms and cylinders: how are cylinders and prisms formed

formed by joining two parallel, identical bases

show shearing nicely with a stack of paper

give the stack of paper a push from the side so that the sheets of paper slide over to understand shearing of solid shapes, think of the thin slices as made of paper

what does the height and the area of the base tell us

height tells us how many layers the prism is made of and the area of the base tells us how many unit cubes are in each layer

why do we multiply the height by the area of the base to calculate the volume of a prism or cylinder? consider a rectangular prism that is 4 units high and has a base of area 6 square units,

if we fill this prism with 1 unit by 1 unit by 1 unit cubes, then the prism will be made of 4 layers. each layer has 6 cubes in it = 1 cube for each square unit of the area in the base. so the whole prism is made of 4 groups of layers with 6 cubes in each group, and therefore there are 4 x 6 cubes in the prism. each cube has a volume of 1 cubic unit, therefore the volume of the prism is 4 x 6 cubic units

ex: what does it mean to say that the volume of a solid shape is 30 cubic centimeters

it means the solid shape could be made without leaving any gaps with a total of 30 1 cm by 1 cm by 1 cm cubes, allowing cubes to be cut apart and pieces to be moved if necessary

if you have water in a container and you pour the water into another container, what happens to the volume of water

it stays the same, even though the shape changes

volue formulas for pyramids and cones: how are pyramids and cones formed

joining a base with an apex ( a point)

as with a shape in a plane, we can shear a solid shape and obtain a new solid shape that has the same volume...explain using a polyhedron

lets start with a polyhedron, pick one of its faces, and then imagine slicing the polyhedron into thin slices that are parallel to the chosen face (rather like slicing a salami with a meat slicer) imagine giving those thin slices a push from the side, so that the chosen side remains in place but so that the other thin slices slide over, remaining parallel to the chosen face and remaining the same distance from the chosen slice throughout the sliding process

what's the most primitive, basic way to determine the volume of a solid shape

make the shape out of unit cubes (filling the inside completely) and to count how many cubes it took

what's the volume of a solid shape

measure of how much three dimensional space the shape take sup

how is the height measured when it comes to prisms or cylinders

measured in the direction perpendicular to the bases, not on the slant

what's the height of a pyramid or cone

perpendicular distance between the apex or the pyramid or cone and the plane containing the base height is measured in the direction perpendicular to the base , not on the slant

what's shearing

process of sliding infinitesimally thin slices

what's the difference between surface area and volume of a solid shape

surface area is a measure of how much paper, cloth, or other thin substance it would take to cover the outer surface of the shape volume of a shape is a measure of how much stuff it would take to fill the shape

what does the cavalier's principle about volume says

that when you shear a solid shape the volume of the original and sheared solids shapes are equal. volume is not added or taken away during shearing, just shifted over

in the volume formula it is understood that if the height is measured in some unit

then the area of the base is measured in square units of the same unit the volume of the pyramid or cone resulting from the formula is then in cubic unis of the same basic unit

in the volume formula it is understood that if the height is measured in some unit...

then the area of the base is measured in square units of the same unit. the volume of the prism or cylinder resulting from the formula is then in cubic units of the same basic unit

where does the fraction 4/3 in the sphere formula come from

there is a way to show how the volume of a half sphere is the same as the volume of a cylinder with a cone removed from it we can use cavalier's principle to explain this, which applies not only to shearing but actually to any case where two solid shapes have the same cross section at every height so to derive the volume of a sphere, we show that a half sphere and a cylinder minus a cone have the same cross section at every height, thus they have the same formula we can find the volume of a cylinder minus a cone because we know the volume formula for cylinders and cones.

explaine how we can determine the volume of a box by multiplying its height times its width times its length

think of the box as subdivided into layers, and each layer as made up of 1 unit by 1 unit by 1 unit cubes. each small cube has volume 1 cubic unit, and the volume of the whole box (in cubic units) is the sum of the volumes of the cubes, which is just the number of cubes

what's the first step in understanding how to determine volumes in other ways

use the moving and additivity principles for volume

what's the formula for volumes pyramids and cones

volume = 1/3 x (height) x (area of base)

the volume formula for a sphere: the volume in cubic units of sphere of radius r units is given by the formula

volume = 4/3piRcubed

what's the formula for volumes of prisms and cylinders

volume = height x area of base

the volume formula for prisms and cylinders is also valid when the height or the area of the base is a fraction or a decimal, explain

we must consider layers that contain partial cubes, such as layers containing 7.5 cubes, and we must consider partial layers..ex, if a prism is 4.5 units high instead of 4 units high, then we would have 4.5 groups of cubes instead of 4 groups of cubes.. so we would have 4 full groups of cubes and another half group of cubes, meaning another group of cubes that has only 1/2 as much volume as the other groups

to understand how to determine volumes and to understand volume formulas, what must students first know

what volume means