Why is the lateral area of a cone is pi*r*sqrt(r^2+h^2)?
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- Опубліковано 10 чер 2024
- Geometry proof! Learn why the lateral area of a cone is pi*r*sqrt(r^2+h^2). This is a must-know topic for your geometry class!
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First impressions.
Lateral area of a cone is a combination of area of a triangle 1/2*b*h, perimeter of a circle for base 2*π*r, and height using hypotenuse of a right angle triangle c=√(a²+b²).
Edit: it is good to see how sectors are used to find the correct proof.
2:40 bro forgot to edit and cut the footage
dont care hes epic
lol I just noticed that haha
I think it's become his schtick at this point
he's down that before. it's fun😅
The amount of dedication sir does to explain us is crrazy. Love from India
Thank you!
I just did it by surface area of a revolution solid, was really good practice
You always keep us engaged in the amazing world of mathematics. Thanks for your contribution
Thank you!
Nice video.
I only knew the calculus based way to prove this and never bothered to learn a purely geometric way to do it so this is pretty cool
Wouldn't it be easier to integrate, adding up circumferences of gradually decreasing values? The radiuses can be evaluated using the function (r/h)(h-x), and then we can integrate this function evaluated at x from 0 to h. The result should be the product of this integral and 2π, giving πrh as the final result.
I'm not sure i did the calculations correctly though, since that would imply l=h
You are right. That will be “calculus basics”. This channel is bprp “math basics” so we just do basic geometry for this.
Btw L and h are different. So be careful.
And in doing so you'll end up with a Jacobian that will give you the coefficients that are here calculated using arc proportions etc. The rest will be (more or less) an integral for area of circle.
Essentially, every time you say "area of circle is π * r^2", you are implicitly integrating in polar coordinates. You've just memorized the result.
What's going on @2:45 lol
That's how sausage is made.
He forgot to edit it out
There's also a fun way of "proving" it with calculus looking at the cone as a limit of regular pyramids. For a regular pyramid it's easy to prove that its lateral area is pl, where p is semiperimeter and l is hight of the faces.
This only works if you believe that this sequence of regular pyramids really "converge" (in some sense) continuesly to a cone (which it does, but it's a whole another pain to prove that it does). Makes great intuition for the formula though.
You should have a look at the final question for the math paper 3 edexcel gcse (the one with the 2 hexagons and circle) i think it would be interesting to see!
plz do video on fourier trasnform
Excelente
I tried it myself first and found the same answer by dividing the pie arc length 2*pi*r by the total circle 2*pi*sqrt(r^2+h^2) and multiplying that with the area of circle pi*r^2*h^2. All that simplifies into the same formula :)
When I saw the formula I understood, it's actually quite simple
You have an underlying assumption that was not addressed. "The lateral area laid flat is a sector of a circle." That needs to be justified. I'm not challenging it, I can prove it to myself. I know that this is nitpicking but if there were ever a subject where nitpicking is the watchword then that subject would be maths.
No, he proved it for this particular solid. How did he do it? He painted over the surface and moved it. The problem with the proof is that only works as a visual proof for this example until he uses calculus over an arc length of a straight line multiplied by the circumference of a circle integrated from 0 to h. That will show it is always true.
Sir why negative times negative gives positive?
Hello, if you would go straight, (we'll call it the positive direction) 2 times, a single step, you would go forward: "1 step times 2", which are 2 positive steps.
If you suddenly, have the amazing urge to walk backwards, then you would walk a step, in the backwards direction, or a "minus step". You would do so 2 times and therefore: -1 * 2 = -2
Now, lets say somebody recorded you the whole time- and they now play the video in reverse! So when you walked forward- in the video, now you are seen walking backwards, and when you walked backwards... Now it will look like you walk forward!
So! When you walk backwards (first negative), and when you are filmed, and the video in reverse (second negative), you are actually doing the same thing as if you just walked forward, as if you just went to the positive direction
Hope that explained it :)) It is me recalling Dr. Aviv Tsenzor's video.
There are many videos on UA-cam answering this.
You can refer video of channel "bhannat maths"
"Turn around, turn around again. Wtf? I'm facing the same direction"
Wise word from wise man
Turn around, Turn around again!
I'm facing the same direction!
I think this video is missing some editing.
Do Americans not use the term "surface area"? Lateral area feels more clunky to me, but I can see places where it would be useful.
the ground circle area is not included , or is it by that term .
It is because “surface area” refers to the entire cone, including the bottom circle. Here we are only interested in finding the area of the curved portion of the cone.
@@Ninja20704 Awesome, thanks so much. I've never seen that term before. This makes sense to me.
let f(x) be a third degree polynomial with a leading coefficient 1. there is no integer k s.t. f(k-1)f(k+1)
The answer is just (p)irl
Area of sector is similar to area of triangle or maybe special case of it
Who agree with me?
calculus
If h -> 0, the formula becomes pi * r * sqrt(r^2) or pi * r^2. At least one degenerate cone confirmed for circle.
All that integration and hard as f problems in the past, and you just end up here 😂
hey round -0.5 to the nearest integer
under common round, -0,5 rounded is 1, cause if the fractional part is exactly .5 you round up by standard. The problem is that if the number has a .5 fractional part, theres two nearest integers, the upper and the lower, so you just choose one depending on what you need
Bprp I love u
I hv just fuc**d my life since 6 months