How did they figure out the area of a circle is πr² ?

How did they figure out the area of a circle is pi times r squared?

0:00 intro
3:10 Archimedes
6:25 Leonardo and Sato Moshun
8:20 RABH

Archimedes
Leonardo and Sato Moshun
Cut The Knot area of a circle
Area of a circle, formula explained (mathematicsonline)
Understanding the Formula for Area of a Circle #shorts (mathematicsonline)
Rabbi Abraham bar Hiyya Hanasi
Area of a Circle by Rabbi Abraham bar Hiyya Hanasi
David Garber and Boaz Tsaban, A mechanical derivation of the area of the sphere, The American Mathematical Monthly 108 (2001), 10--15.
Wikipedia

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[larzcaetano]

I value much more this type of content and historical background investigation than solving nearly-impossible questions/puzzles. Amazing video!

[FaneBenMezd]

Fun fact, the letter π was chosen to describe the ratio of circumference/diameter because of the word 'περιφέρεια' (~periphery) that is the 'περίμετρος' (perimeter) of a circle.

[jimspelman]

Just for fun, I pulled out my old Calculus book from college and integrated using the equation for a circle (after solving for y to get y= sqrt(r^2-x^2). I simply integrated from 0-1 (quadrant 1 of a circle with radius r with its center at (0, 0)) and multiplied by 4. Sure enough...Area still equals (pi)r^2 !

[pinedelgado]

Thank you!! I've just done it!! Upon learning of William Jones 1706 work you referenced, I went to Wikipedia and found out that this work (with the first use of the Greek letter pi for representing the ratio between a circle's circumference and its diameter) is titled "Synopsis Palmariorum Matheseos." Then within mere minutes, I found myself buying a copy of it on Amazon!! Thank you so much, Presh, for enlightening me!!!!

[keshavmtech]

Wow, thank you so much for enlightening on very fundamentals of formula for a Circle's area.

[georgeray]

Math history is amazing. This was excellent! Thank you.

[chrismoule]

r² is the area of a square implied by two radii of a circle of radius r that are at 90° to each other.

4r² is therefore the area of the square in which the circle with radius r is inscribed.

If πr² is the area of the inscribed circle, then π/4 is the fraction of a square with sides 2r that is taken up by its inscribed circle.

Maths is a beautiful thing.

[fun-damentals]

archimede's method was so good. felt like a plot twist. thats why i love maths

[wildrice]

Great video, Presh. Love your videos, but this one is one of my favorites ... really well explained, thank you!

[mtmdesigns]

I like this format as an addition to the usual problems.

[jojoonyoutube]

The animations in this video were very good, Presh!

[dr.johnslab]

I love how excited you get, Presh! Keep it up! 👏👏👏

[i-mosh]

This is just amazing. Getting to know this now after all these years

[hippophile]

I wondered about this very thing years ago. With the relatively modern benefit of calculus you can prove the relation easily enough, but fascinating to see the ancient arguments which somehow I missed out on. Thanks!!

[Bokery]

Great video! It's fascinating to see that they came to such an understanding 2, 000 years ago.

[JMan]

Love this history of math stuff. I know middle school and high school students don't really care about where math comes from (I didn't), but I do think it's vital to teach it before college. You never know whose interest could pique.

[BillionFires]

Here is a simple method: Picture the concentric rings from the third approach. The innermost ring has a circumference of zero (it's basically a point). The outer ring has a circumference of 2πr. Now add up all the circumferences by integrating 2πr from 0 to r. It's a very simple integral that results in πr^2

[MateusScheffer]

Hey, I'm a brazilian fan of your channel, and I loved it last year when you solve our Pinocheo math problem from OBMEP. This year we again had an interesting problem about some flowers which, if you want to, I can translate to you.

[pietergeerkens]

If we consider a square (the base shape from which we define the measurement of area) with half-side of length r, then the area of said square is Aₛ = 4⋅r². Then the area of a circle inscribed in that square is
π⋅r² = π⋅(4/4)⋅r²
= (π/4) ⋅ 4⋅r²

= ¼π ⋅ Aₛ
and we see that ¼π is the "form factor" for area of a circle in the same way that:
- ½ is the area form factor for a triangle: Aₜ = ½ ⋅ bh;
- ⅓ is the volume form factor for a cone or pyramid: V_cone = Vₚ = V_cone = ⅓ ⋅ A_base ⋅ h; and
- ⅔ is the volume form factor for a hemisphere: Vₕₑₘᵢₛₚₕₑᵣₑ = ⅔ ⋅ A_base * h = ⅔ ⋅ π⋅r² ⋅ r = ⅔⋅π⋅r³.

[gregorymccoy]

I really like methods 2 and 3. This was a great video!