The document describes how to calculate the area of a circle with a diameter of 2 feet using the standard formula of A = πr^2 as well as an ancient Egyptian method. The Egyptian method involves drawing an octagon inside the circle and calculating its area, which turns out to be approximately 7/9 of the area of the square enclosing the circle, equivalent to 3.11 square feet compared to the exact value of 3.14 square feet using the pi-based formula. This Egyptian method produces a result that is remarkably close to the correct value, with only about a 1% error.

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

2. 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. 2 feet

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

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

5. Click your mouse for the next idea ! The Egyptian Octagon Method 2 feet Finally... we draw lines to divide the small squares in the corners in half, cutting them on their diagonals. 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

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

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

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

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 !!