2 feet

How would you calculate the area of this circle ?

...probably using the formula A = R2

Since the diameter is 2 feet,

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The constant , called “pi”, is about 3.14

so A = R2 3.14 * 1 * 1 3.14 square feet

means “about equal to”

?R

1 foot

“R”, the radius, is 1 foot.

2 feet

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?

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

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?

The Egyptian Octagon MethodThe Egyptian Octagon Method

Draw a square around the circle just touching it at four

points.

What is the AREA of this square ?

2 fe

et

Well.... it measures 2 by 2, so the

area = 4 square feet.

2 feet

Click your mouse for the next idea !The Egyptian Octagon MethodThe Egyptian Octagon Method

2 fe

et

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

Click your mouse for the next idea !The Egyptian Octagon MethodThe Egyptian Octagon Method

2 fe

et

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

Click your mouse for the next idea !The Egyptian Octagon MethodThe Egyptian Octagon Method

2 fe

et

The EGYPTIANS were very handy at finding the area of this Octagon

19

After all, THIS little square has an area 1/9th of the big one...

19

19

19

19

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. 18

1. 18

1. 18

2 feet

Click your mouse for the next idea !The Egyptian Octagon MethodThe Egyptian Octagon Method

2 fe

et

...and ALTOGETHER we’ve got...

1. 18

1. 18

1. 18

1. 18

4 pieces that are 1/18th or 4/18ths which is 2/9ths1

9

19

19

19

19

Plus 5 more 1/9ths

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

square

2 feet

Click your mouse for the next idea !The Egyptian Octagon MethodThe Egyptian Octagon Method

2 fe

et

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.

79

AMAZINGLY CLOSEAMAZINGLY 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 !!about a 1% error !!