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

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2 feet

“R”, the radius, is 1 foot.

R

1 foot

so A =  R2

3.14 * 1 * 1

 3.14 square feet

2 feet

 means “about equal to”

Click your mouse for the next idea !

How would you calculate the area of this circle ?

...probably using the formula A = R2

Since the diameter is 2 feet,

?

The constant , called “pi”, is about 3.14


2 feet

2 feet

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

2 feet

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

2 feet

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

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.

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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

1

9

1.

18

1.

18

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

1

9

1

9

1

9

And so do these four others...

1.

18

And each corner piece is 1/2 of 1/9 or 1/18th of the big one

1.

18

1

9

2 feet

Click your mouse for the next idea !

The Egyptian Octagon Method

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

2 feet


2 feet

1

9

1.

18

1.

18

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

1

9

1

9

1

9

Plus 5 more 1/9ths

1.

18

1.

18

1

9

2 feet

Click your mouse for the next idea !

The Egyptian Octagon Method

...and ALTOGETHER we’ve got...

2 feet

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


2 feet

We have an OCTAGON with an area = 7/9 of the original square.

7

9

2 feet

Click your mouse for the next idea !

The Egyptian Octagon Method

FINALLY...

Yep, we’re almost done !

The original square had an area of 4 square feet.

2 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

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 !!


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