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Discovering Circle Areas: The Egyptian Octagon Method Explained

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Explore the fascinating approach that ancient Egyptians used to calculate the area of a circle through the Egyptian Octagon Method. This method begins by inscribing a square around the circle, then dividing it into nine smaller squares and transforming it into an octagon. Learn how this ancient technique remarkably approximates the area of the circle, yielding results very close to modern calculations using π. Join us as we delve into the intersection of history and geometry, revealing how ancient methods continue to amaze with their accuracy.

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Discovering Circle Areas: The Egyptian Octagon Method Explained

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  1. “R”, the radius, is 1 foot. R 1 foot so A =  R2 3.14 * 1 * 1  3.14 square feet 2 feet  means “about equal to” Click your mouse for the next idea ! How would you calculate the area of this circle ? ...probably using the formula A = R2 Since the diameter is 2 feet, ? The constant , called “pi”, is about 3.14

  2. 2 feet Click your mouse for the next idea ! LETS explore how people figured out circle areas before all this  business ? The ancient Egyptians had a fascinating method that produces answers remarkably close to the formula using pi. ?

  3. 2 feet Click your mouse for the next idea ! The Egyptian Octagon Method Draw a square around the circle just touching it at four points. ? 2 feet What is the AREA of this square ? Well.... it measures 2 by 2, so the area = 4 square feet.

  4. 2 feet Click your mouse for the next idea ! The Egyptian Octagon Method Now we divide the square into nine equal smaller squares. Sort of like a tic-tac-toe game ! 2 feet Notice that each small square is 1/9 the area of the large one -- we’ll use that fact later !

  5. 2 feet Click your mouse for the next idea ! The Egyptian Octagon Method Finally... we draw lines to divide the small squares in the corners in half, cutting them on their diagonals. 2 feet Notice the 8-sided shape, an octagon, we have created ! Notice, also, that its area looks pretty close to that of our circle !

  6. 1 9 1. 18 1. 18 After all, THIS little square has an area 1/9th of the big one... 1 9 1 9 1 9 And so do these four others... 1. 18 And each corner piece is 1/2 of 1/9 or 1/18th of the big one 1. 18 1 9 2 feet Click your mouse for the next idea ! The Egyptian Octagon Method The EGYPTIANS were very handy at finding the area of this Octagon 2 feet

  7. 1 9 1. 18 1. 18 4 pieces that are 1/18th or 4/18ths which is 2/9ths 1 9 1 9 1 9 Plus 5 more 1/9ths 1. 18 1. 18 1 9 2 feet Click your mouse for the next idea ! The Egyptian Octagon Method ...and ALTOGETHER we’ve got... 2 feet For a total area that is 7/9ths of our original big square

  8. We have an OCTAGON with an area = 7/9 of the original square. 7 9 2 feet Click your mouse for the next idea ! The Egyptian Octagon Method FINALLY... Yep, we’re almost done ! The original square had an area of 4 square feet. 2 feet So the OCTAGON’s area must be 7/9 x 4 or 28/9 or 3 and 1/9 or about 3.11 square feet

  9. AMAZINGLY CLOSE to the pi-based “modern” calculation for the circle ! 3.11 square feet 3.14 square feet only about 0.03 off... about a 1% error !!

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