Transcript Calculate Pi Dean

A few years ago, there
was this guy named
Archimedes.
A few years ago, there
was this guy named
Archimedes.
He was one of the first
people to calculate pi.
A few years ago, there
was this guy named
Archimedes.
He was one of the first
people to calculate pi.
He didn’t have a
calculator.
Or, mathematically
speaking ….
Or, mathematically
speaking ….
Capture the circle
with an outside
square…
Capture the circle
with an outside
square.
Pretend the red
square is a poorly
drawn circle.
Pretend the red
square is a poorly
drawn circle.
Let’s calculate pi
for the red square.
Pretend the red
square is a poorly
drawn circle.
Let’s calculate pi
for the red square.
It will be a poor
estimate of pi.
Remember how to
calculate pi?
Remember how to
calculate pi?
Remember how to
calculate pi?
1
Let’s say the radius
of the circle is 1.
Remember how to
calculate pi?
1
2
Let’s say the radius
of the circle is 1.
This means each side
will be 2.
Remember how to
calculate pi?
2
Let’s say the radius
of the circle is 1.
This means each side
will be 2.
Circumference = 2
Remember how to
calculate pi?
2
2
Let’s say the radius
of the circle is 1.
This means each side
will be 2.
Circumference = 2 + 2
Remember how to
calculate pi?
2
2
2
Let’s say the radius
of the circle is 1.
This means each side
will be 2.
Circumference = 2 + 2 +2
Remember how to
calculate pi?
2
2
2
Let’s say the radius
of the circle is 1.
This means each side
will be 2.
2
Circumference = 2 + 2 +2 + 2
=8
Remember how to
calculate pi?
2
Circumference = 8
The diameter of the
square will simply be
the distance across.
It is 2.
Remember how to
calculate pi?
2
The red square
version of pi is now:
pi = 8/2= 4
Circumference = 8
Diameter = 2
This is our first
estimate of pi.
It is 4.
2
Circumference = 8
Diameter = 2
This is our first
estimate of pi.
It is 4.
2
Circumference = 8
Diameter = 2
That’s not very close
because our red
square wasn’t a very
good circle.
This is our first
estimate of pi.
It is 4.
2
Circumference = 8
Diameter = 2
That’s not very close
because our red
square wasn’t a very
good circle.
Next, let’s draw a
better one.
Let’s do it again with
a hexagon.
Let’s do it again with
a hexagon.
A hexagon is a better
circle.
Let’s do it again with
a hexagon.
A hexagon is a better
circle.
It should be a better
estimate of pi.
Radius of 1 again.
1
Let’s take a second to learn
how to get the perimeter of
a hexagon if you don’t
know how long one side is.
This distance is called the
apothem.
apothem
This distance is called the
apothem.
apothem
We can find the perimeter if
we know this distance.
apothem
Your calculator has a button
called tangent. You will
learn what this means in
Trigonometry. For now, we
are just going to use the
button on the calculator.
Your calculator has a button
called tangent
apothem
You will learn what this
means in Trigonometry.
Your calculator has a button
called tangent
apothem
You will learn what this
means in Trigonometry.
For now, we are just going
to use the button on the
calculator.
Your calculator has a button
called tangent
apothem
You will learn what this
means in Trigonometry.
For now, we are just going
to use the button on the
calculator.
Find the calculator button
called TAN
Here is the formula for
perimeter if you know only
the apothem:
apothem
Here is the formula for
perimeter if you know only
the apothem:
apothem
Perimeter = (2n)tan(180/n)
Here is the formula for
perimeter if you know only
the apothem:
apothem
Perimeter = (2n)tan(180/n)
n = number of sides
Let’s say this apothem is 1.
1 apothem
Let’s say this apothem is 1.
n=6
1 apothem
Let’s say this apothem is 1.
n=6
1 apothem
Per = (2*6)tan(180/6)
Let’s say this apothem is 1.
n=6
1 apothem
Per = (2*6)tan(180/6)
= 6.9282
Radius of 1 again.
1
Radius of 1 again.
1
With our new formula, we can
get the perimeter of the
hexagon.
Radius of 1 again.
1
Circumference = 6.93
With our new formula, we can
get the perimeter of the
hexagon.
Per = (2*6)tan(180/6) = 6.93
Hex circumference = 6.93
Hex diameter = 2
2
Circumference = 6.93
Diameter = 2
Hex circumference = 6.93
Hex diameter = 2
2
Pi = 6.93/2 = 3.47
Circumference = 6.93
Diameter = 2
This is our second estimate of
pi.
It is 3.47
This is our second estimate of
pi.
It is 3.47
Notice that it is closer than the
square’s estimate of 4.
Let’s do it again with an
octagon!
Let’s do it again with an
octagon!
Look at how well an octagon
simulates a circle.
Let’s do it again with an
octagon!
Look at how well an octagon
simulates a circle.
This estimate should be pretty
good. Better than the other
two anyway.
Radius of 1 again.
1
Radius of 1 again.
1
An octagon has 8 sides, so
Per = (2*8)tan(180/8) = 6.63
Radius of 1 again.
Oct circumference = 6.63
2
Circumference = 6.63
Diameter = 2
Oct diameter = 2
Radius of 1 again.
Oct circumference = 6.63
2
Oct diameter = 2
Oct pi = 6.63/2 = 3.32
Circumference = 6.63
Diameter = 2
Look at the pattern of how
things are going…..
Look at the pattern of how
things are going…..
Square pi = 4
Look at the pattern of how
things are going…..
Square pi = 4
Hex pi = 3.47
Look at the pattern of how
things are going…..
Square pi = 4
Hex pi = 3.47
Oct pi = 3.32
Look at the pattern of how
things are going…..
Square pi = 4
Hex pi = 3.47
Oct pi = 3.32
The more sides on our
polygon, the closer we get to
pi.
What if we had 100 sides?
100 sides is very close to a
circle. It is difficult to draw,
so just imagine the red
polygon has 100 sides.
What if we had 100 sides?
100 sides is very close to a
circle. It is difficult to draw,
so just imagine the red
polygon has 100 sides.
This 100-gon pi should be
even closer to pi.
Radius of 1 again.
1
Radius of 1 again.
1
100-gon circumference =
(2*100)tan(180/100) = 6.2853
Circumference = 6.2853
Radius of 1 again.
100-gon circumference = 6.2853
2
Circumference = 6.2853
Diameter = 2
100-gon diameter = 2
Radius of 1 again.
100-gon circumference = 6.2853
2
100-gon diameter = 2
100-gon pi = 6.2853/2 = 3.1426
Circumference = 6.2853
Diameter = 2
Radius of 1 again.
100-gon circumference = 6.2853
100-gon diameter = 2
100-gon pi = 6.2853/2 = 3.1426
Our 100-gon has the first 4 digits
correct!
You can now find as many places
of pi as you would like.
You can now find as many places
of pi as you would like.
The more sides your polygon has,
the more decimals of pi you can
derive.
Just think how close to a circle a
billion-gon will look.
Just think how close to a circle a
billion-gon will look.
All you have to do is find the
circumference.
Just think how close to a circle a
billion-gon will look.
2
All you have to do is find the
circumference.
The diameter will always be 2.
Let’s go ahead and find the
circumference of a billion-gon?
Let’s go ahead and find the
circumference of a billion-gon?
Per = (2*1,000,000,000)tan(180/1,000,000,000)
Let’s go ahead and find the
circumference of a billion-gon?
Per = (2*1,000,000,000)tan(180/1,000,000,000)
= 6.2831853071795864975961378867589
Let’s go ahead and find the
circumference of a billion-gon?
2
Per = (2*1,000,000,000)tan(180/1,000,000,000)
= 6.2831853071795864975961378867589
The diameter is still 2.
Let’s go ahead and find the
circumference of a billion-gon?
2
Per = (2*1,000,000,000)tan(180/1,000,000,000)
= 6.2831853071795864975961378867589
The diameter is still 2.
Pi= 6.2831853071795864975961378867589 / 2
Let’s go ahead and find the
circumference of a billion-gon?
2
Per = (2*1,000,000,000)tan(180/1,000,000,000)
= 6.2831853071795864975961378867589
The diameter is still 2.
Pi= 6.2831853071795864975961378867589 / 2
= 3.1415926535897932487980689433794
So, here’s that billion-gon:
Billion-gon circumference is
2
2 x 1 x 1,000,000,000 x tan(180/1,000,000,000)
= 6.2831853071795864975961378867589
Billion-gon diameter is 2.
Billion-gon pi = 6.2831853071795864975961378867589 / 2
= 3.1415926535897932487980689433794
Billion-gon is correct to the 16th decimal!!
Get as many places as you can. Just
keep adding more sides to your
polygon and you will get a better
and better estimate of exact pi.
= 3.1415926535897932487980689433794
Billion-gon is correct to the 16th decimal!!