How did Archimedes Calculate Pi?

How did Archimedes manage to determine the value of pi without calculus?

Connect with us at:

This video was made in association with The Math Centre at Humber College, by Sharmin Kassam.
Producer - Cameron Redsell-Montgomerie

[TheChilibeano]

Actually sin, cos, tan were not around during this time. He used harmonic and geometric means to find the inner and outer perimeters. Just saying...

[slashine]

Remember, this person was born in 300 B.C

[stephenwatchesyoutube]

i’ve seen so many different stories about how archimedes did this that i’m convinced no one really knows exactly. people just reverse engineered the formula and made up their own story

[ffggddss]

In the practical sense of actually *calculating* π, let alone how Archimedes did it, this presentation provides nothing.

[acorn]

0:32, there should've been an explanation of why we're talking about a triangle when we were just talking about squares.

[hughbo]

Wow, intellectual knowledge spewed faster than my brain could mop it up.

[abbasmehdi]

I wanna know that what is that pi? Actually, why pi is there? Why is the area of circle pi times (radius)^2? I want the derivation of area or perimeter of circle?

[volin]

Let me try to explain this better.
1) Construct a circle of radius R and circumferance C
2) Draw a square enclosing the circle (one that intersects at 4 points) with a side 2R
The perimeter of the enclosing square is 8R and thus C<8R

3) Draw an internal square of the circle intersecting it at 4 points.
4) Find a triangle connecting a point at the end of one of internal square's sides, the corresponding mid-point and its centre (this is a 1/8 triangle of the square)
5) From the right angle theorem we know that the squared hypotenuse equals the sum of the other two squared sides (c^2=a^2+b^2). Since we know that the shorter sides are equal, we can solve for R^2=a^2+a^2 -> R^2=2a^2 -> sqrt(R^2/2). so the complete side of the internal square is 2*sqrt(R^2/2).
6) from this we can ascertain an approximate relationship between circumferance and diameter of a circle:
8*sqrt(R^2/2) < C < 8R

[justinhansen]

0:53 A way to get that triangle using special triangles: it's a special triangle so sides 1 1 and a hypotenuse of square roots of 2. To go from the square root of 2 to one times by the square root of 2 to get 2 and divide by 2 to get one. That means we can times each side by square root of 2 over 2 and get your triangle

[gauraangsonkar]

So you are saying Archimedes was familiar with sine and cosine Functions ?
Which was Developed by Aryabhatta and brahmgupt almost ~750 years after "Archimedes" ? (according to popular accepted dates)?
on the other hand he can draw a few circles and measure it different ropes and just average it out?

[gyphryphobia]

how do people understand this without understanding how they have arrived at the jargon or degrees of angles etc etc etc

[ekoi]

π=C/d
inscribe an n-sided polygon inside the circle
n=numbe of sides of polygon
s=length of side of polygon
divide the polygon in n equal parts giving you n equal sectional triangles
let θ=angle (top of the isosceles triangle) of a sectional triangle
θ=360°/n because if you divide 360° by n will give you the angle of the sectional triangle
cut the isosceles triangle from its tip to its base
θ/2=(360°/n)/2=180°/n


d/2 because in the sectional triangle the two legs are actually the radius of the polygon or the circle which is half the diameter

sin(θ/2)=s/d

consider the perimeter of the polygon is the circumference of the circle
C=ns
s=C/n

θ/2=180°/n

sin(180°/n)=s/d

substituting s in sin(θ/2)=s/d
sin(180°/n)=(C/n)/d
n sin(180°/n)=C/d
C/d=π

π=n sin(180°/n)

as n approaches ∞,  π=3.141592653589793238462643383279502884197169399375105820974944592307816406286 ...

[sarvajagannadhareddy]

How to find THE INDIAN Pi ?
Draw a circle of radius 0.5. Draw another radius at 90 degrees. Join 2 ends. We get chord 0.707106781...Half of it is 0.35355339...Take 7 radii equal to 3.5 and subtract 0.35355339.. and we get 3.146446609...It is Circumference and also Pi (14 - root2)/4.7 Radii --- Half Chord = CIRCUMFERENCE (Pi).Truth is a roaring fire. --- SJR

[spectre]

Damn you archimedes you make our life more harder than we thought

[PrabhatKumar-yvlj]

But if we doesn't know the value of pi then how can we find the formulae of circumstance?

[guyintears]

So pi is a circumference of a circle with diameter 1?

[krishanuchatterjee]

Hey, it was Indian mathematician Aryabhatya who gave the value of pi. As 3.1415

[gregkowalik]

Where does opposite equals √2/2 come from?

[criticalcreator]

Beautiful. I do wonder if my math teacher knows that. He's very technical and less into history. I should introduce him to this channel, simple, to the point, quite understandable as well as interesting.

[goldyolvidobaroa]

This is what happens when you don't browse on Facebook too much 😅😆