1. Why π ? Do you know why we must use 3.14 in all area and circumference formulas?

2. 2 feet How would you calculate the area of this circle ?...probably using the formula A =  r 2 Since the diameter is 2 feet, The constant , called “pi”, is about 3.14 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.

3. 2 feet ? 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 π.

4. Ancient Egyptians

5. Egyptians today!

6. 2 feet ? 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.

7. 2 feet 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 !

8. 2 feet 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 !

9. 2 feet The Egyptian Octagon Method 2 feet The EGYPTIANS were very handy at finding the area of this Octagon 1919 After all, THIS little square has an area 1/9 th of the big one... 1919 1919 1919 1919 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

10. 2 feet The Egyptian Octagon Method 2 feet...and ALTOGETHER we’ve got... 1. 18 1. 18 1. 18 1. 18 4 pieces that are 1/18 th or 4/18 ths which is 2/9 ths 1919 1919 1919 1919 1919 Plus 5 more 1/9 ths For a total area that is 7/9 ths of our original big square

11. 2 feet 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 We have an OCTAGON with an area = 7/9 of the original square. 7979

12. AMAZINGLY CLOSE to the pi-based “modern” calculation for the circle ! 3.11 square feet3.14 square feet only about 0.03 off... about a 1% error !!

13. ? feet Your Turn…… ? feet It’s your turn to discover pi, π using the octagon method! Get into groups of 3 solve the problem given to each group. Remember, you need the diameter!

14. Class group work ……. Group # Octagon Method A=s² if D= ? Estimate of π using Octagon Method 7/9 * s² A = π r² #17 / 9 timesft² #27 / 9 timesft² #37 / 9 timesft² #47 / 9 timesft² #57 / 9 timesft² #67 / 9 timesft² #77 / 9 timesft² #87 / 9 timesft² #97 / 9 timesft²

15. Using π………. Irrational number… 3.14159265…. Continues forever….. Never repeats a pattern or a single digit. A = πr² Area formula of a circle! If the diameter is 10 in. What is the radius?