The Length of the Nile River… Oh that’s Just the beginning says FERMI!

The opportunity that Fermi questions provide to students is extraordinary.  These big real world math problems provide a fantastic platform to hone mathematical skills and allow students to show and practice their understanding of mathematical concepts.  The variety of the problems is incredible, and one can be made up from just about anything due to the vast amount of data available today.  I look forward to seeing how I can incorporate these expansive problems into my units as a way to challenge students’ reasoning and problem solving abilities.

The Fermi problem I explored as an exercise was to determine the length of the Nile River divided by the average width of a strand of human hair?  The length of the Nile River seemed like a great starting point for my work in the problem.  One point that students often struggle on is breaking word problems down into easy to understand components.  The beginning of the question specifically states the length of the Nile River.  So I thought, does the problem require length in kilometers, miles, meters, or some other unit, or could we use any of these?  Also, which would be easiest to use?  The word “divided” immediately follows the first portion, so I thought division.  The last portion of the question is perhaps the most difficult to understand.  The word “average” stuck out to me, and my thinking was that this portion was critical to break down as it was a problem in itself.  The average or mean is the calculated “central” value of a set of numbers, or in this case, the set of all widths of the strands of human hair.

I would guess the length of the Nile river is about 7,000 kilometers long.  I think this because I know the United States including the width of Alaska additionally is about 4200 miles wide, and I thought that the Nile river was about the same length as the distance from one side of the United States to the other.  So 4200 miles x 1.6 kilometers per mile= 6700 miles is about the length of the Nile river.

For the average width of a strand of human hair I had no idea, but I guessed it was about two tenths of a millimeter.  I later would look up both of these values to do the problem, but those were my guesses.  Most Fermi problems have you estimate the values like these, but I wanted to have a more precise answer.

Nile River part of length

The length of the Nile river I found in kilometers via Google, then converted to millimeters so that it would be easier to divide by the average centimeters of a human hair strands width.  With a quick google, I found that the length of the Nile river was 6,853 kilometers long.  To convert this length to millimeters I multiplied:

6853 kilometers x 1,000,000 millimeters/kilometer= 6,853,000,000 millimeters for the length of the Nile river.

For the average width of a strand of hair it was not so easy to find.  When I looked up the width, I was presented with the diameter of a human hair.  This caused me to think, is this truly the value I need?  Since a strand of hair is shaped kind of like a cylinder the diameter is equivalent to the width of a strand of human hair.  This portion of the problem requires critical thinking skills and a thorough understanding of geometric figures.

Hair Strand Diameter

According to the Physics fact book, the average diameter of a strand of human hair is 100 micrometers.  To convert micrometers to millimeters, I had to multiply:

100 micrometers x 1millimeters/1000micrometers= .1 millimeters is the average width of a strand of human hair.

The last portion of this problem required taking the length of the Nile river 6,853,000,000 millimeters

Divided by the average width of a strand of human hair .1 millimeters.  So

6,853,000,000 millimeters  ÷ .1 millimeters= 68,530,000,000 hair strands or 6.853 x 10^10.  The millimeters in the units cancel leaving just the hair strands.

When I checked my answer with the Fermi questions answers I saw that there answer was 66 billion hair strands .  The difference in our answers may have come from there using a slightly different or more precise value for the average width of a strand of human hair.  There value must have been slightly larger.  Probably there average width of a strand of human hair was 68,530,000,000 /66,000,000,000=1.038,

so there average width of a strand of human hair was .1 millimeters x 1.038 times larger=.1038 millimeters.

Though I was slightly off, I was pretty excited by how close my answer was to the actual answer and found the problem to be very rewarding because it required me to explore real world objects, science, math and even geography.  FERMI problems have major potential for the future of mathematics because they require lots of exploration and challenge the understanding of mathematics for a student in a fun, unique way!

FERMI= Math Future and Present



  1. First of all, I’m impressed with your determination! A lot of people (me included) would be intimidated by the sheer size of the numbers you worked with. I think this could be a really good exercise for students who are nervous about large numbers. There’s isn’t a set answer with this question, which I think might alleviate some of the pressure of getting the “wrong answer.” It could be a fun activity in which students could compare answers and maybe laugh a bit along the way about the ridiculous nature of the problem itself. I think questions like this have the potential to inspire students to use math to solve other things (like distances in space, if they are so inclined to use large numbers) on their own. Many people believe math is not – and cannot be – creative, and this would be an excellent way to show the creative nature of math. Good job!


  2. How many heads of hair is that? More or less than the number of people to measure the length of the Nile?

    Not a typical Fermi problem because of the use of given information. How long would you have guessed the Nile to be and how wide a hair? Fermi would say estimate them both would result on average in a decent answer. Also, given the givens, why do you think your answer might have been different by a factor of 10? (66 billion = 6.6×10^10) (content)

    I’m glad these excited you, though, and this is a fun post.
    clear, coherent, complete, consolidated: +


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