Aquarium of the Pacific's "Blue Cavern"	Volumes of Space Figures
	When done with this lesson, you will have demonstrated how to find the volume of prisms, cylinders, pyramids, and cones given dimensions. find a dimension of a prism, cylinder, pyramid, or cone when given the volume and other dimensions. explain how changes in one dimension affect the volume of a prism, cylinder, pyramid, or cone.
	Approximate completion time: 3 hours

"Volume—that space which is filled up."

Volume measures the space in 3D that an object occupies. As such, three dimensions must always be included in measuring volume.
Consider the following formulas for volume. Do you see how three dimensions are included?

Shape	Picture	Volume
Rectangular Prism		V = l·w·h
Triangular Prism		V = ½·a·b·h
Cylinder		V = π·r2·h
Rectangular Based Pyramid		V = 1/3·l·w·h
Cone		V = 1/3·π·r2·h

You must think critically to see the three dimensions in the cylinder and cone's formula.
Do you see the squaring of the radius? Two dimensions are included in r² since it really is r·r.

 

Although many people love to remember formulas, you can find volumes with some simple principles of thinking.

Volume Type #1

"Stacking Congruent 2D Shapes"
	The volume of a solid formed by sweeping a 2D area through a 3rd dimension is simply V = B·h where B is the area of the base (the 2D area that was "stacked") and h is the height of the solid (the distance between the end bases).

 

Volume Type #2

"Stacking Congruent 2D Shapes that diminish to a point "
	The volume of a solid formed by "vanishing" a 2D area through a 3rd dimension is simply V = 1/3·B·h where B is the area of the base (the 2D area that was "stacked") and h is the height of the solid (the distance between the base and the vertex).

 

Therefore, the volume of a cones, prisms, cylinders, and pyramids—whatever the shape of
the base—can be found by using either V = B·h or V = 1/3·B·h.

 

Now it's your turn to practice applying volume formulas and see what happens when dimensions are tweaked.
Try your hand at some practice and some deep thinking problems in Assignment 8.03.