MATH 113: Volume and Surface Area
In this activity you will calculate the surface area and volume of four different prisms. You will then compare these values to learn how shape and dimensions affect sizes.
Work in groups of 4; each student should turn in a single paper. Each group should have 4 sheets of 8.5 by 11 inch paper and some tape or paper clips.
Take an 8.5 by 11 inch piece of paper and roll it longways into a tall, narrow cylinder (circular prism) that is 11 inches tall and has a circumference of 8.5 inches. Tape it together; use paperclips to hold it together if you have no tape.
Now, roll an 8.5 by 11 inch piece of paper into a short, wide cylinder with a height of 8.5 inches and a circumference of 11 inches. Tape it together.
- Compare the two cylinders. Which do you think has the larger volume? Why?
- Which do you think has the larger surface area? Why?
| ||Tall Cylinder||Wide Cylinder
||Tall Prism||Wide Prism|
|| || || || |
|| || || || |
- Compute the volume of the tall, narrow cylinder. Show your work here and enter the result in the table above.
- Compute the surface area of the tall, narrow cylinder. Show your work here and enter the result in the table.
- On a separate sheet of paper, compute the volume and surface area of the short, wide cylinder. Record your results in the table.
- According to your calculations, which cylinder had the greatest volume?
- Which had the greatest surface area?
- Given the choice between increasing the height of a cylinder by 10% or increasing its circumference by 10%, which will result in the greatest change in volume? Explain your answer. (If you're not sure, complete the worksheet and return to this question. Does it help to answer the question for prisms first?)
Take a piece of paper and fold it in half length-wise, and then in half again. The paper will be divided into four tall, narrow quarters. Fold and fasten the paper so that it (approximately) makes a tall, narrow square prism.
Take a piece of paper and fold it in half width-wise and then in half again. Make a short, wide square prism with the sheet of paper.
Bonus: If you cut a piece of paper in half and tape the two halves together, you can make much taller and narrower or shorter and wider prisms. Suppose you did this. Which would have the greater volume -- a very, very tall narrow prism, a very, very wide short prism, or a prism with moderate height and width? Justify your answer.
- Which prism do you think has the greater volume? Why?
- Which prism do you think has the greater surface area? Why?
- Calculate the surface area and volume of the short, wide prism. Show your work below and enter the result in the table.
- Calculate the volume and surface area of the tall, narrow prism. Enter the results in the table.
- Which prism had the greatest volume? Which had the greatest surface area?
- Compare your answer for square prisms to your answers for circular prisms (cylinders). In general, which do you think has greater volume, a short prism with a large circumference or a tall prism with a small circumference? Explain your reasoning.
- In both examples, the tall prism had less surface area than the short one. Can you explain why this has to be true?