Easy as Pi

During a wonderful visit to the Galvin Middle School in Wakefield, we noticed signs indicating that the entire school would be celebrating pi day next week on March 14 -- that is, 3.14. Apparently the entire school will be wearing PI-rate shirts that day as well. Seeing a few of them today reminded me of a real pirate story from 2009, involving one of my friends and his coffee. I posted it on one of my other blogs with the title Aarrrrr-abica!

More importantly, we realized that we had brought a round object to a school that was about to celebrate pi, so I posed a few questions about it and promised to post some more.

EarthView has a flat bottom (so that it does not roll around like a hamster ball). That bottom is a circle 12 feet in diameter, containing the continent of Antarctica and the Southern Ocean. If not for the flat bottom, EarthView would be a sphere 22 feet in diameter. For the questions below, it is probably best to imagine it as a perfect sphere. With these two dimensions -- and some formulas that might be in the back of your math book or at 1728.com -- you can calculate answers to the following. Be sure to keep track of your measurement units!

• What is the circumference of the flat, bottom part of EarthView?
• What is the area of the flat, bottom part of EarthView?
• If 24 people are in EarthView, how much space does each person have to stand?
• What is the surface area of EarthView (assuming it is a sphere) in square feet? What is the surface area in square yards?
• What is the volume of air in EarthView?
• If it takes six minutes to fill EarthView with our trusty fan, what is the rate of air flow in cubic feet per minute?
• If we filled EarthView with water, how much would it weigh? (For this you need to look up or calculate the density of water in pounds per cubic foot.)

I mentioned to some classes that the Christian Science Church in Boston has a stained-glass globe 30 feet in diameter, called the Mapparium. Located in the Back Bay area near the Prudential Center, it represents the political boundaries of the entire world in 1935. Admission is $6 for adults, $4 for students, and free for MTA members. It is a great family outing that offers a great comparison with EarthView!

Given the Mapparium's 30-foot diameter, use pi to answer the following questions:

• What is the surface area of the Mapparium?
• What is the volume of the Mapparium?
• How much bigger is the surface area of the Mapparium, compared to EarthView?
• How much greater is the volume of Mapparium than that of  EarthView?

The real earth is approximately 8,000 miles in diameter. (In reality, it is not quite spherical, and it is not exactly 8,000 miles in any case, but this rounded number is suitable for the calculations below.) What is its diameter in kilometers? Given this information, use pi to answer the following:

• What is the surface area of the earth in square miles? In square kilometers?
• What is the volume of the earth in cubic miles? In cubic kilometers?

Finally, consider your classroom globe. Use a string to calculate its circumference of the globe, and then use pi to find its diameter in inches. Use pi to find:

• The surface area of the globe in square inches and in square feet.
• The volume of the globe in cubic inches and cubic feet.

What patterns can you discern among all of the numbers you have calculated? How do changes in diameter, area, and volume relate to each other as spheres of different sizes are compared?