Volume of pyramids intuition | Solid geometry | High school geometry | Khan Academy

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The volume of a pyramid is a fraction of the volume of the rectangular prism that encloses it. We can find out what that fraction is by cutting a prism into several pyramids.
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[@LimitlessMathLLC]

Always great content! I love that you are explaining the WHY!!! I’ve always been inspired by your channel. I’ve admired your videos for many years and finally got the courage to start my own math channel.

[@SerenitySong6]

This is the easiest explanation I've seen, doesn't deal with series and all that crap. Thanks!

[@amandad422]

Love this channel.Doing Geometry in class currently,I'm getting MOST of it,but the videos You have REALLY help a lot.Thank You so much.

[@yiyingteo]

Thank you for answering the WHY Great content! Love it! Would like to see another episode explaining abt the volume of a cone

[@Scott_G]

These are so great. I wish Sal was making videos back when I was in school.

[@ukaszszczur5696]

How do you know that the left, right, front and back pyramids have also k in their equations? How do you know that these constants of proportionality are equal?

[@haimanotmengistu9793]

The constant is the same for all pyramids since we are using it as part of the formula and not as a constant for that specific prism out of which it was cut

[@ghxst.5128]

he uses the same argument as he did for the original pyrimid that he wanted to find the volum of

[@youareloved6778]

Thankyou for these Math concepts! I’m really understanding!

[@saksham1919]

Even when i am taking precalc, calc, etc. you can still teach a dog some new tricks. Thank you so much for these videos. Keep it up

[@vasilymorozov]

Really happy to see another video from khan academy about math. I actually have a video on my channel about word problems using volume!

[@usualavantgasp]

It is fascinating to know how neat je draw each pyramid sketch using DIGITAL PEN

[@anishprasai9398]

Love❤ from Nepal🇳🇵 Grateful for your videos 😃

[@philipmcclure6493]

I have a question. Why is the constant K the same between the different colored pyramids? Wouldn’t each pyramid have a different constant K?

[@SoumilSahu]

Think of it this way: the area of a circle is pi*r^2. The size of the circle is irrelevant. The area depends on just r and the pi sitting out front stays unchanged. Similarly, the volume of the pyramid just depends on ITS x, y and z

[@070279381]

@@SoumilSahu doesn't mean we can assume the constant will be the same. it can be kxyz + mxyz + nxyz= xyz it just assumption from Sal to make k = m = n

[@thesardaunatv]

Amazing video my mentor I am following your footsteps And I need your blessing to prosper

[@thesardaunatv]

Thank you sir in advance

[@the_butter_god7487]

Ye

[@jadonjung5074]

I was confused about how to get the volume of pyramids but your video helped me @Khan Academy

[@khalilulahnabil5574]

excellent explanation ❤️❤️❤️

[@Infinitelybetter466]

Hey khan, I appreciate the video you uploaded. My question is did you drive the one-third because in an essence there are dimensions to the pyramid? So one-third times the three sides which in this you called xyz. Thanks

[@sarbjeetsingh1111]

Wonderful.

[@070279381]

this is with assumption the constant is the same k for each cases.

[@santiagocos4769]

Gorgeous.

[@jasoncampbell1464]

It's a bit funny how other commenters are so satisfied with the explanation and find it easy. From my perspective this is really clever and unobvious. It's not intuitive at all that you should make the assumption that the ratio k between the volume of the pyramid and the box that contains it will be the same for every pyramid. And even with that assumption it's not easy to foresee the usefulness, in finding a general formula, of expressing the volume of a box of arbitrary dimensions in terms of the 6 pyramids. You would need the intuition that the pyramids, despite differing dimensions, have the same volume. That's not obvious at all. At the end of the video I still don't see the grounds for making that assumption except to try it out and see that it works, coincidentally. Finding the formula itself is easy with calculus but it's helpful to develop these geometric intuitions edit: I think I understand why someone would find this assumption useful now. Fundamentally the assumption is that the volume of the pyramid is *linearly proportional* to the volume of box that contains it. So if you've done enough algebra, you'll know that expressing the volume of the box as a sum of the volumes of pyramids will yield a linear equation in which x,y,z will cancel out and you can solve for the constant of proportionality. It's more of anticipating a linear equation where x,y,z cancel than having the intuition that the inner pyramids have the same volume (it's impossible to tell that a-priori). That's a pretty good game plan.

[@MohoganyFan]

I have no idea why I am watching this...perhaps I'll use it in the future

[@georgeorourke7156]

Very elegant presentation but I wonder whether one doesn't need an argument to ensure that k is somehow not dependent on the dimensions of the pyramid.

[@coopshopdesigns4890]

Thank you

[@maxwell4546]

A lovely pyramid scheme you have there.

[@Jokeland3172]

Good 😍😍😍

[@username-yn5yo]

Does anybody know what programm he is using as a chalkboard?

[@Raynover]

SmoothDraw

[@bluejeanswhiteshirt9481]

This is how many people use khan academy because of corona 👇👇👇

[@biplobhimu]

Ah😮❤22-10-2023

[@YoshaGang]

....

[@deshn21]

Are you Khan?

[@PitOfDepression]

No he is sal

[@unknownjah6920]

Deleted your video

[@PitOfDepression]

What?

[@duzhenfkhrodole3703]

Your advertyis disturb and iam very angry of your app cut it shut up

[@markgriz]

¯\_(ツ)_/¯

[@hayleysmith347]

what tf does this even mean