Post on 05-Jan-2016
Why π?
Do you know why we must use 3.14 in all area and circumference formulas?
2 feet
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
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
?
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.
Ancient Egyptians
Egyptians today!
2 feet
?
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
The Egyptian Octagon MethodThe Egyptian Octagon Method2
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 !
2 feet
The Egyptian Octagon MethodThe 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 !
2 feet
The Egyptian Octagon MethodThe Egyptian Octagon Method2
feet
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
The Egyptian Octagon MethodThe Egyptian Octagon Method2
feet
...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
The Egyptian Octagon MethodThe Egyptian Octagon Method2
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.
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 !!
? feet
Your Turn……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!
Class group work …….
Group #
Octagon Method
A=s² if D= ?
Estimate of π using Octagon Method
7/9 * s²
A = π r²
#1 7 / 9 times ft²
#2 7 / 9 times ft²
#3 7 / 9 times ft²
#4 7 / 9 times ft²
#5 7 / 9 times ft²
#6 7 / 9 times ft²
#7 7 / 9 times ft²
#8 7 / 9 times ft²
#9 7 / 9 times ft²
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?