Volume of a Sphere, and a Soccer Ball

When I think about spheres I first think of soccer balls.  Not just any soccer ball, but a jabulani world cup ball!

Ball used at the World Cup in South Africa

As a student four years removed from Calculus, I first think about the volume of a sphere in terms of the basic volume formula.  I am excited to use this math formula because I will likely be teaching about the formula in a 7th grade mathematics classroom I am assistant teaching at this year.  One key thing to note about formulas like the volume formula is that they were not always so readily available.  I was curious where the formula

V=4/3*pi*r^3

Actually came from: Archimedes is said to have discovered and proved the formula during the 3rd century B.C. in Greece.  Of all his works the work he is said to have been most proud of was the discovery of the relationship between a sphere’s volume and a cylinder circumscribing that sphere.  This is particularly impressive considering he also invented a claw that destroyed ships, a very close estimation of pie, and created a place value system of base 100,000 because he was disappointed with the greek number system.  He was so proud of this discovery that he even had a sculpture of the phenomena on his grave.  The relationship between the two is that the a sphere with the same diameter as the height and diameter of a cylinder has 2/3 of the volume of a cylinder.

Archimedes found that a sphere’s volume is 2/3 that of a circumscribing cylinder of the same height and diameter.

Moving past Archimedes, about 2000 years later Gottfried Leibniz and Isaac Newton worked separately to discover Calculus.  How can two men discover calculus at the same time in separate places? Check here

Calculus can be utilized to prove the formula for the volume of a sphere.  I could spend a lot of time in my own words explaining the proof for the volume of a sphere utilizing Calculus, but rather than waste your time I will point you to two helpful sites that explain the process.  The first is a webpage that shows how integration and the pythagorean theorem can be used to show that the Volume of a sphere is what Archimedes said it was.  The second is a video that walks learners through the proof.

What is incredible and fascinating about the proofs for the volume of a sphere is that Archimedes proved it without any Calculus! Calculus had been no where near invented in the 3rd century, but Archimedes still was able to form a geometric proof.  In fact, the proof is so challenging and complex that most textbooks have a simplified version.

Returning back to the beginning, a Jabulani soccer ball is a sphere.  I wondered what the volume of that sphere is?  It turns out the answer is not as straight forward as I first thought.  There is not an instant plug and chug as there so rarely is in real life.  As I searched for the specifications of a Jabulani match soccer ball.  I could find only the circumference which is considered to be between 68.5 cm and 69.5 cm.  So I knew there would be some error in my answer as the circumference was a range.  By first utilizing the formula for Circumference I had to find the radius of the sphere.  That formula is:

Then I had to carry out some algebra to solve for the radius.  So I divided 69 centimeters by 2*pi.  When I did this I found the radius of a jabulani soccer ball to be about 11 centimeters.     I inserted the radius into the formula for the volume of a sphere.

and found that the volume of a jabulani soccer ball was 5546.44 cubic centimeters.  If I wanted to further challenge myself or my students I could ask them the range of the volume of all FIFA regulation jabulani soccer balls.

So what should we take from this exploration through the history of finding the volume of a sphere, and the volume of a jabulani soccer ball?

First, the history of mathematics is fascinating and may be something that teachers can use to engage students in a lesson, and help them to realize why mathematics matters!  Another key discovery is that mathematical problems do not often give us the necessary information to solve them in one step.  We must seek out and find a way to figure out the information we are missing to find the solutions we are looking for.  Lastly, there are so many mathematical resources at our disposal today that educators need to carefully consider the problems that they are giving students.  A good question to sometimes ask students may be, “Are math problems in real life this simple? Why do you think that?” This practice will help students in their critical thinking skills and their ability to recognize the many variables that influence a real problem.  Math can be challenging, but students will benefit much more from our teaching if we challenge them and expect their best effort!

1. ramseybr says:

I like how you bring in the historical aspects when you talk about the volume of the sphere. Showing the history behind mathematics, as you mentioned, helps people understand that mathematical relationships, like the one between the volume and the cylinder, are discovered and require problem solving skills. Also, the links to other resources are great for those who want to delve deeper into the calculus and proof behind the formula.

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2. ramseybr says:

correction: between the volume of the sphere and the cylinder….

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3. Nick Karavas says:

Kevin, your posts never cease to amaze me. I would say that this “application” of mathematics was one of the most straight forward and useful posts that I have seen even past our class. I like the idea of challenging students with deeper questions such as, “will life be as easy as this?” It seems as if our teachers always tell us that we will need mathematics to survive in the real world yet the mathematics that we compute within middle school and high school is no where near what we will accomplish in real life. Thus, it is important to bring in the research part of mathematics to show where we find our answers to questions that we have. Within the above post you have made multiple connections that overlap in some way or another which I thought was a terrific use of the post. Nice job Kevin.

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4. More world cup soccer ball maths: http://archive.bridgesmathart.org/2015/bridges2015-151.pdf
(OK to to say maths because football.)

I like the emphasis in this post on the story of math, and the subtext of different methods. Maybe include a couple other of Archimedes inventions so we know how impressed he was, when he says that this is his favorite. For a class activity, the summary is a perfect fit: after calculation, find the actual volume by immersion.

5C’s +

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