circle area proof

8/7/2019 Circle Area Proof

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2 feet

How would you calculatethe area of this circle ?

...probably using the formula A = π R 2

Since the diameter is 2 feet,

Click your mouse for thenext idea !

The constantπ

, called“pi”, is about 3.14

so A = π R 2 ≈ 3.14 * 1 * 1

≈ 3.14 square feet

≈ means “about equal

to”

? R 1 foot

“R”, the radius, is 1 foot.



2 feet

Click your mouse for the next idea !

?

LETS explore how people figured outcircle areas before all this π business ?

The ancientEgyptians had a

fascinating methodthat produces

answers remarkablyclose to the formula

using pi.



2 feet

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?

The Egyptian Octagon MethodThe Egyptian Octagon Method

Draw a square around thecircle just touching it at four

points.

What is the AREA of thissquare ?

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

Now we divide the squareinto nine equal smaller squares.

Sort of like a tic-tac-toegame !

Notice that each small

square is 1/9 the area of the large one -- we’ll usethat fact later !



2 feet


Finally... we draw lines todivide the small squares inthe corners in half, cuttingthem on their diagonals.

Notice the 8-sided shape,an octagon, we havecreated !

Notice, also, that its arealooks pretty close to that of

our circle !



2 feet


The EGYPTIANS were very handy atfinding the area of this Octagon

19

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

19

1

9

19

19

And so do these four others...

And each corner piece is1/2 of 1/9 or 1/18 th of the big

one

1.18

1.18

1.18

1.18



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 ths19

1

9

19

19

19

Plus 5 more 1/9 ths

For a total area that is7/9 ths of our original big

square



2 feet


FINALLY... Yep, we’re almost done !

The original square had an areaof 4 square feet.

So the OCTAGON’s areamust be 7/9 x 4 or 28/9

or 3 and 1/9

or about 3.11 square feet

We have an OCTAGON with anarea = 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 !!

circle area proof

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