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

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

The Egyptian Octagon Method 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. 2 feet

The Egyptian Octagon Method 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 ! 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 ! 2 feet

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

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

The Egyptian Octagon Method Click your mouse for the next idea ! The Egyptian Octagon Method FINALLY... Yep, we’re almost done ! We have an OCTAGON with an area = 7/9 of the original square. 7 9 2 feet The original square had an area of 4 square 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 2 feet

to the pi-based “modern” calculation for the circle ! 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 !!