1. ? How would you calculate the area of this circle ?
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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”

2. 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. ?

3. The Egyptian Octagon Method
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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

4. The Egyptian Octagon Method
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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

5. 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

6. The Egyptian Octagon Method
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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

7. For a total area that is 7/9ths of our original big square
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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

8. The Egyptian Octagon Method
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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

9. 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 !!