Volume of Rectangular Prisms, Cubes, and Platonic and Archimedean Solids

Every student is taught the formulas for the volume of a rectangular prism as length times width times height, the three dimensions of a rectangular prism. The answer must be give in cubic units. A volume of12 (for sides of 2x3x2), is meaningless because this suggests a one dimensional distance of 12 units when we are talking about 12 cubic units.

Of course, a cube is a rectangular prism but all three dimensions are equal so we can simplify the formula by saying side cubed = side x side x side. I have asked my student to master the first 12 cubes and to try this without a calculator if they can do multiplication or with a calculator if they cannot. Remember, multiplication is just repeated addition, so 3 cubed or 3 x 3 x 3 is just 3 + 3 + 3 = 9, then 9 + 9 + 9 = 27.

It is fun to look at page two of the pdf where I gave the 12 rubiks cubes and they can physically see the cubic units. I really interesting cube is 12 cubed since that gives you the number of cubic inches in one cubic foot of 1728 cubic inches. 

On pages 3 and 4 of the pdf, I ask the children to take any rectangular prism in their house. A tissue box is my example, because it was the closest prism to me when I created the lesson. Notice that I estimated the dimensions by rounding to the nearest whole inch. They should do small prisms in cubic inches and large prisms like their room or their house by cubic feet. My steps are three feet each but a child’s is usually between 1.5 and 2.5 feet so they can use this or they can use a ruler or a tape measure. Heights of ceilings may be more difficult by a guideline is that most ceiling heights are between 7 and 12 feet. 

Another fun exercise is to convert cubic feet to cubic inches by multiplying the cubic feet by 1728.

My box of 510 cubes cost me $50. So if they figured out that their room would require 1,000,000 cubic inches, they could divide by 510 and multiply by $50 to figure out the cost of filling the room with foam cubic inches; in this case $98,039.

For the 3-5th graders, get your calculator (iphone, ipad, computer or just a calculator with a square root button). Following the rules for order of operations (parentheses, exponents, multiplication and division, addition and subtraction), figure out the volume of each Platonic solid and Archimedean Solid. By the way, they will never do this exercise unless they become math majors in college. Remember, the solids we used in class each had an edge of 3 inches, so each formula requires that you multiply by 3 cubed or 27. If they build one of these solids with magna tiles or other forms, they will need to measure that edge and multiply the formula by the cube of that edge. For example, if I build a solid with an edge of 125 after calculating the formula I would multiply it by 125 or 5 cubed.

Have fun and please do not try to fill your room with these cubic inches. The cost is prohibitive.