Capture that volume and surface area

Recall, volume is the amount of cubic units or space that an object occupies in 3 dimensions. Surface area is the combined area of all of the faces (it is a 2-dimensional construct).

1. Given the cubes, find the following:

a. The volume of one cube in cubic inches______

b. The volume of one cube in cubic centimeters______

c. The surface area of one cube in square inches______

d. The surface area of one cube in square cm______

2. Using your cubes, make the assumption that the side length is 1 inch.

Construct any rectangular prism with 12 cubes.

a. What is the volume of this prism?______

b. What is the volume of this prism in cubic inches?______

c. How did you find your answer to #2a,b? Briefly explain.

d. Will all of your “classmates” construct the same prism? Why or why not?

e. Sketch your prism. Look at another group’s prism and provide a sketch of it.

f. What is the surface area of your prism? ______

g. What is the surface area of their prism? ______

h. Do all prisms with the same volume have the same surface area? Why or why not?

Exploration

You are given 12 cubes to package together in any way you choose. You want to minimize the packaging required. You only need packaging for the parts of the cube exposed to air.

1) What is the minimum packaging required? Sketch the configuration of cubes that will require the least amount of packaging and justify your answer.

2) If you are given one more cube so that you need 13 cubes packaged, what is the minimum packaging?

3) If you are given two more cubes, three more cubes, four more cubes,…

4) If you are given n more cubes, what is the minimum packaging?

Exploration

Using your cubes, make the assumption that the side length is 1 unit.

1) Take your building block to be one cube. What is the volume?

2) Take your building block to be a cube that has the twice the length of the original building block. What is the volume, surface area?

3) Take your building block to be a cube that has three times the length of the original building block. What is the volume, surface area?

4) Fill out the following table:

Side length / N
Area of face
Surface area
Volume

5) Use your table to answer the following:

a. When the side length is doubled, the volume is______.

b. When the side length is doubled the surface area is______.

c. When you multiply the side length by N, the volume is multiplied by ______.

d. When you multiply the side length by N, the surface area is multiplied by ______.

e. Prove statements 5c, 5d by algebraic methods.

6) In the explorations above, you were working with rectangular prism(s) (a cube is a particular rectangular prism). Do the same relationships between side length, volume and surface area hold for other solids?

7) Consider a sphere. Does it make sense to say “double the side”? What could we change about a sphere?

8) Consider a cylinder. Does it make sense to say “double the side”? What could we change about a cylinder?