Q308a, Arc length and Area of Sector of a circle

Proof of an arc length of a circle,
Proof of an area of a sector of a circle,
☀️ Patreon: / blackpenredpen

[@yossefbudagov8748]

Finally you are doing some geometry. I'm looking forward for more geometry and trigonometry in this channel :)

[@PranitSuman]

OMG YOU ARE THE BEST MATHS TEACHERRR

[@borg972]

2:30 Where did you get a whiteboard with autocomplete? I want one too

[@blackpenredpen]

Magic! : )

[@zmaj12321]

1 month ago...??

[@eduardomamede6548]

@@zmaj12321 seems that we found a glitch in the matrix

[@concerneddoge9013]

I love this video, short and precise. Do more of this revisiting of math basics.

[@alicechapman2513]

All the other online sites have much more complicated proofs than these, this was the only one I could understand! Thank you :))

[@zmaj12321]

Your valuable sacrifice to the Integral War has not been forgotten, soldier!

[@Kmrabhinav569]

Also, for arc length, for very small value of theta, r and d(theta) will form a triangle with length of r*d(theta). Integrating that wrt theta will give us r* theta which is the final answer

[@krishpanjta2121]

Dude I found your way of teaching amazing

[@IcaroCamposdeAPinto]

Fun fact: If yo take the integral of r*theta, it's actually equal to r^2*theta/2

[@danielsergio8531]

Thats not a coincidence if you think about it

[@esthergetsy5496]

clearly explained. Thank you sir.

[@bipinmaurya3824]

Nice sir... Sir make a vedio on plane projection

[@DrewAsWellAs]

Nice to see 10 grade math again :)

[@vincentespe3602]

merry christmas Blackpenredpen :)

[@blackpenredpen]

Thank you, you too!!

[@vincentespe3602]

+blackpenredpen hello idol.. :).. frm philippines.

[@blackpenredpen]

: )))

[@kindpotato]

He makes videos with all this crazy math and then he just does this.

[@blackpenredpen]

Yes.

[@MarioFanGamer659]

That 1/2 in the sectoral area remidns of the Tau-enemies' counterargument: The formula for a circle's area doesn't look as pretty with tau because you have got otherwise fractions (π * r² vs. τ/2 * r²). But you if you take into consideration that the arc length is r * θ and the sectoral area 1/2 * r² * θ then the 1/2 exists there for a reason.

[@denisdailyjohndo636]

Quite good

[@xaxuser5033]

0 dislikes at the moment i hope so the next 24 h

[@edrodriguez5116]

Im not giving up on this math!!!!!!!!!

[@blackpenredpen]

Merry Xmas Eddie!

[@edrodriguez5116]

Likewise professor !

[@lapischicken]

Seems like this video was uploaded a month ago, set to unlisted/private, and then set to public again

[@blackpenredpen]

Yup!

[@harshranjan8526]

Good shot Riots 😁

[@adityaaryan1610]

Seeing your channel's name I didn't expect you to use a blue pen

[@ZoneEEEEEEEEEEEE]

You might want to check the thumbnail once again

[@blackpenredpen]

I fixed it, thanks!

[@JoséAntonioBottino]

At minute 2:27 you say that “this is when you have radians”. You do not actually place θ rad and 2𝜋 rad as you do for θ° and 360°, and you are omitting the unit radian (rad) which many consider a dimensionless unit. The variable θ is the number of radians and not the angle measure in radians. Here is the explanation of the formula: Let s denote the length of an arc of a circle whose radius measures r. If the arc subtends an angle measuring β = n°, we can pose a rule of three: 360° _______ 2 • 𝜋 • r n° _______ s Then s = (n° / 360°) • 2 • 𝜋 • r If β = 180° (which means that n = 180, the number of degrees), then s = (180° / 360°) • 2 • 𝜋 • r The units "degrees" cancel out and the result is s = (1 / 2) • 2 • 𝜋 • r that is, half of the circumference 2 • 𝜋 • r s = 𝜋 • r If the arc subtends an angle measuring β = θ rad, we can pose a rule of three: 2 • 𝜋 rad _______ 2 • 𝜋 • r θ rad _______ s Then s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r If β = 𝜋 rad (which means that θ = 𝜋, the number of radians), then s = (𝜋 rad / 2 • 𝜋 rad) • 2 • 𝜋 • r The units "radians" cancel out and the result is s = (1 / 2) • 2 • 𝜋 • r that is, half of the circumference 2 • 𝜋 • r s = 𝜋 • r If we take the formula with the angles measured in radians, we can simplify s = (θ rad / 2 • 𝜋 rad) • 2 • 𝜋 • r s = θ • r where θ denotes the number of radians (it does not have the unit "rad"). θ = β / (1 rad) and θ is a dimensionless variable [rad/rad = 1]. However, many consider θ to denote the measure of the angle and for the example believe that θ = 𝜋 rad and radians*meter results in meters rad • m = m Mathematics and Physics textbooks state that s = θ • r and then θ = s / r It seems that this formula led to the error of believing that 1 rad = 1 m/m = 1 and that the radian is a dimensionless derived unit as it appears in the International System of Units (SI), when in reality θ = 1 m/m = 1 and knowing θ the angle measures β = 1 rad. In the formula s = θ • r the variable θ is a dimensionless variable, it is a number without units, it is the number of radians. When confusing what θ represents in the formula, some mistakes are made in Physics in the units of certain quantities, such as angular speed. My guess is that actually the angular speed ω is not measured in rad/s but in (rad/rad)/s = 1/s = s^(-1).

[@notnilc2107]

Merry Christmas

[@harshranjan8526]

thank you same to you

[@aaronthielemann1617]

why is the area equal to the integral of the arc length?

[@friedkeenan]

You can think of a bunch of arcs being stacked on top of each other until it reaches the outside of the circle. Now imagine each arc had a bit of area to it. The area for an arc a distance r away from the center with an angle of θ would be rθ*dr. So let's add them all up. Let's say R is the radius of the circle, so the sum of all those tiny areas would be the integral from 0 to R of rθdr. That equals .5R^2*θ-.5*0^2*θ which simplifies to .5R^2*θ

[@davidbrisbane7206]

@@friedkeenan Very clever 😁.

[@aaronthielemann1617]

@@friedkeenan That is a very helpful answer thank you very much!

[@bleppss2769]

Adding on to this why is the circumference equal to the derivative of the area

[@adrgonzato]

how lovely

[@rome8726]

Good video. I have a question concerning the series. How can i prove that the coefficient of the binomial series (1+4x)^(1/2) are integers ?

[@erikdurfey5576]

Intro = awesome

[@blackpenredpen]

Thanks!

[@akkimylo3286]

It may be a stupid question but, how can we just assume that the ratio of the arc lengths to the perimeter or sector area to circle area is the ratio of the θ/2π? Shouldn't we have to prove that, even though we can all see that it's true?

[@snuffles7222]

Very Fun!

[@vikashkumarsingh6515]

Thank you sir. Jai Hind.

[@Anonymous-kw7ls]

awesome

[@Hacker118]

i am interested in the covered word

[@Whatever-sy7hv]

circumfercen gyazo.com/a5e1d99362763d464267069a41647a22

[@blackpenredpen]

OMG, LOL

[@Hacker118]

@@blackpenredpen what a fail lol

[@PeterNjeim]

@@blackpenredpen During the last few frames the overlay was removed from video, that's how @Whatever got the image. If you really want to hide your mistakes, leave the overlay on until the very last frame. If you still want the video to look good, make the fade to white at the end of the video *also* apply to the overlay. I saw in the video that the word Circumference didn't fade to white, making it fade to white would make the video still look clean, while still covering your mistake. Merry Christmas!

[@blue_blue-1]

Like this intro.

[@emontrailers]

did you record the music from an elevator

[@alejandrofabiangarcia5917]

I THOUGT YOU WERE GOING TO USE CALCULOUS! HAHA

[@blue_blue-1]

Alejandro Fabian Garcia, it is „calculoose“.

[@alejandrofabiangarcia5917]

@@blue_blue-1 really? 😂 😂 😂 Sorry my natal language is spanish xd

[@blue_blue-1]

Alejandro Fabian Garcia, No, calculus! (A comma matters).

[@peterciccone620]

is this why when you're working with hyperbolas the area is 1/2 of the angle?

[@pavsjoga2271]

U've mentioned Black pen and red pen, but where the heck the blue pen comes from...

[@tarunparia8981]

Here two pi are not same. Then how can we eliminate bith?

[@JD-ee4df]

There's a mistake in the thumbnail of the video. Good video otherwise! 👍

[@hugodeuxcentsoixante-seize2261]

It was a clickbait : i couldnt get why it was the formula in the thumbnail and not the one in the video so i clicked

[@dangerfar]

@@hugodeuxcentsoixante-seize2261 Same

[@blackpenredpen]

It was just a mistake..

[@denisdailyjohndo636]

Idiot

[@fernando3061]

I want that jacket

[@Jacob-uy8ox]

Joyeux Noël!!

[@blue_blue-1]

So there are 3 x-mas presents. One from each king/math-magician/astronomer. ... sadly I don‘t celebrate x-mas, because it is supposedly the wrong date for HIS birth. But all the best to everyone out there!

[@sreenandhan2544]

S=r*sin(theta) for small theta.. S=r*theta....

[@Quwertyn007]

Isn't the first one circular reasoning? At least for r=1

[@chemsonbro7325]

He even used blue pen 😂🤣

[@VndNvwYvvSvv]

"Weyweyw". Use the chenlu!

[@lapischicken]

this is a new level of thumbnail clickbait

[@blackpenredpen]

: )))))))

[@ssdd9911]

u spelled circle as cirlce i also want to know how u spelled circumference

[@lapischicken]

Commented... 1 month ago? Teach me your ways

[@krishpanjta2121]

Dude go on method not spellings

[@ZAYEDAL-GUMAEI]

ليش م ياخذ مكرفون جيب احسن له والله من ااكورة اللي ب ايدة حسبي الله 🙂🙂