Friday, May 20, 2022

Payne Geometry – Pyramids Vs Prisms

So we are recording today's uh class experience yeah it's fun so let's take a look we're in unit 9 and now we are talking about prisms and pyramids and things like that so let's go dive into today's examples uh first things first we're talking about three dimensional figures .

Okay 3d figures no longer flat stuff but technically uh 3d stuff so the first thing that is going to happen is if we were to take a 3d figure such as this particular uh prism i think you don't want to get that flatten like i know like this should be your .

Alley because of you know your your educational prowess remember children hi ah you can see on your stuff uh 3d figures include prisms and remember prisms and these this is actually not not exactly a prism it's technically a 3d .

Oblong cylinder this is a right cylinder it's all kinds of stuff 3d stuff things that you can touch and smooth and and cones and and dodecahedrons and and and tetrahedrons and so many hydrogens like these you gotta love it uh and so this is what we're talking about .

Now first things first about 3d figures if we were to take a 3d figure and basically flatten it on a flat surface we will create what we call a net a net all right so what we're going to do is discuss with this net situation and as soon as i have my notes pulled up .

For me we will discuss it it'll be great yep so a net is let me go ahead and type it down and here comes our first notes for the day by definition a net is a two-dimensional drawing of a three dimensional object long story short okay so and it's going to show .

Uh bases faces edges and vertices okay so show bases faces edges and vertices of the 3d object if i were to take a 3d object flatten it and such what would .

It be so let me let me let me let me let me let me see um so in this particular case we have this example uh oh that didn't really move it alone there you go i know people are still writing so we have this example right here i'll just move that to the side i'll .

Make this a little bit easier there we go one more time this is it so we have a net now we imagine that there was a three-dimensional figure that was at once uh one point available now what do you imagine if you remember from eighth grade seventh grade what was the 3d figure that was .

Flattened let's see if you can pull it out of thin air if you remember what it was called it was what shape let's start with the basic shape this is a square is this a triangle rectangular prism so technically in a way it looks a little something .

Like uh this and this is where i get to get my um my drawing credits back and maybe be impressive to the children am i impressive yet no yes no yay so at one point it was a 3d figure that looked like so and each one of these sides is really represented .

By uh this net for example uh the top side it can be this that matches there and so from there let's say this side right here is here uh this a side represents here this face is this face and then way over here is representing there this is unfolding the rectangular prism .

And flattening it out on a flat surface so this is a rectangular prism now number one this is what's very important as we go forward you can name your 3d figures by their basic shape like i said this is a rectangular prism which would make you think that there's .

Got to be a rectangle looking somewhere yes someone say yes yes all right thank you it's not a triangle it's not a cone it's not a a triangular prism it's a rectangular prism okay so uh we're going to use that and kind of move along into some other .

Things we're going to draw a net that would form a cube all right so let me show you uh what that looks like so uh here we go so we want to do a net that would form a cube now my question is how many sides does a cube have so if i had a regular cube .

Uh it would just let's see the cube is a square so there's that and then there is i'll use a different color because i'm colorful like that here we go and then i got uh here we go connect the dots and stuff mr payne all right so how many sides does a cube have .

24 sides are you sure i only count one two three four five six yeah i mean really just like yeah six it's even labeled here it's even labeled see for the people at home let's do one two three four four five six there we go alrighty so uh cubes have six sides so just for the .

Sake of you know remembering from you know eighth grade cubes have six sides now what i want to do is i want to it's currently looking like it has three dimensions now for the sake of uh everybody at home and online there are three dimensions that are going on three dimensions .

Uh what are those height width width okay and length and depth or really so there's there's three kinds of dimensions or length width and depth height base height and depth so that means it's going wide it's going tall and it goes deep .

All right so that we can see there's on on this 2d flat surface on your screen it looks like something so i'm going to draw it flat all right so here's one example i'll do that this is when i get to be an artist aha i became a geometry teacher just so i can start drawing shapes .

Do i have six sides yes all right let's i mean if i'm counting one two three four five and six if i were to fold it all up together would it form a cube all right so then here's another question what about uh this .

Cube let's see i'll do it like this maybe like that and then oh thank you all right so then we'll do it like that and then i'll do it like that all right is that a cube you sure well actually yeah would that fold up into a cube i have some yeses i have some news .

Anyone on the on the online community want to chime in would it be a cube yes or no thank you you've been helpful uh the the actual answer in this case would be no why would it be no why would this not form a cue it has six sides i mean there's one .

There's two here's three four five six what's up yes this once it folds up this is going to be in the exact same position as this which means this is both like the top and i need a bottom over here so i need something to be like shown differently um so perhaps maybe if i had uh so let's say we still have this i'll .

Do it like this would this watch this would this be a cube no why one two three four five how about would this be a cube no no no it's seven sides one two three four five six seven all right hold on would this be a cube yes once you .

Fold it together you would be able to create a cube long story short keep this in mind i might have you do a extra credit assignment that has nets and cubes and things like that about you folding your pieces of paper and creating 3d figures so uh yes this would create a tube so .

This is why that would be very important it helps show all of the dimensions and allows us to draw 3d shapes on a two-dimensional dilio all right so for example we can use this to represent uh two major things that we're talking .

About for the next couple of days prisms and pyramids okay prisms and pyramids but we're going to use this with nets all right um let's keep going all right so you want to draw this that'll be a great idea we're going to determine what kind of 3d shape will be .

Formed by folding each net all right so uh if i folded along these dotted lines what would this first example be i love it try angular because you see these triangular pieces right here now these triangles really help remind us of what it is .

And it's a prism that is correct it is a triangular yeah prism that is a net of a triangular prism what about this one this is a net of a what a pyramid of some sort but what kind do you really huh a pyramid that also pyramids are triangles technically so what else .

What other shape do you see here you just said that so what else what shape you see thank you presumably a square and you said pyramid remember our pyramids we look at the base yeah and that we're going to write down in our notes just to remind us because this will be something that we .

See many times as we just determine what is what so this is technically all review from seventh and eighth grade mathematic geometry stuff but always good to have in our notes as we move forward so now there are some labels that i .

Would like to do uh to kind of help us out so because we have the nets and they represent kind of the 3d figures so on the right is the same thing of what is on the left and so we're going to help kind of see where pieces line up so that way even if i'm showing something you on the screen .

You know what i'm talking about um so i'll just start right here where this thing is and this is called a lactor all edge and it's referring to the fact that really and i'll try to kind of highlight it for you is that this edge goes along the side as it goes up on the uh triangular .

Triangle that's actually a pyramid this is actually a pyramid so this is a square pyramid okay so this is a square pyramid so this is the matching side that i'm talking about right here so it's a lateral edge now remember the word lateral means side remember that nathan yeah equilateral .

Means what nathan equal sides that's correct all right so then uh then also we have the following uh you have a lateral face lateral face so get used to these names as we say them uh then we have what will be the called .

The slant height and i'll explain that in a second you also have the lateral edge over here just by the way then you have of course the base this is all about the base so let's talk about the slant height now remember although this looks flat here this will .

Be folded up to turn into this sort of representation of the 3d figures so really the same slant height that you see on this flat course please i'll i'll just kind of turn this green or something so this is that right here that is also representing right here .

As as if i walk along the side of the pyramid and really as you notice it's a pyramid it'll be slanting in towards the middle you asked a question i'm giving you the answer i hope you heard it and so it'll be slanting towards the middle so it's not the real height .

So the part that is pointing to the real height of this particular pyramid is right here in the middle yes it goes from the base so if you need to write yourself a note remember this goes from the center of the base all right height .

For the actual pyramid goes from the center of the base the slant height goes from the sides uh the edge of the side or the base uh edge to the vertices of this lateral face okay so i'm putting a lot of vocabulary words in your uh in your notes today so here once again is your slant height .

Represented right there and at the very top we'll call that the vertex of your 3d figures so if you're writing your notes and labeling all these pieces you'll hear me saying this throughout the rest of the lessons in unit 9 so get used to these words the .

Slant height is not the same as the height of the pyramid or the prism it is the height of the lateral face and so i'll let you guys get that in your notes so we have all this labeled and one thing that you will see is later on .

In your in these calculations we're going to start taking some slices of the pyramid kind of slicing this out and pulling it out this way and will represent it like this so it was already a 90 degree angle and then this was made with the slant height .

And so that slant height is not the height the height will represent with h but the slant height we're going to represent with the cursive l okay so if you need to write yourself a reminder that height will be represented with h but the slant height .

Will be represented with cursive l you will see that in your formulas as you go through this unit so i wanted to kind of put this in your in your notes now so that when you're asking what is the slant height you are all ready to know that that is represented by cursive l .

Or for some way to say this is the breast cancer ribbon awareness and when is that month was that april october october i never know uh everyone wears pink everyone wears pink but i'm gonna pick all the times i know i wear pink it's possible anyone should know .

Legit all right so just keeping that in our notes and reminders for that okay so let's get into some similarities and differences between the prisms and the pyramids we need to know the difference so prisms versus pyramids now we're going to make a little bit of a list and so i would suggest you do the same first .

Things first something that's very similar now both of these are named for their base okay both pyramids and prisms are named for their base so whatever their base is if it's a triangular base there'll be a triangular prism if those triangular base will be trying to print .

It uh also they have lateral faces lateral basis which also in including lateral edges as well and that's going to be equal to the number of sides of the base okay so the lateral faces and edges are going to be equal to the number of sides .

Of the base so if there are four sides of the base then there will be four lateral faces so for example if it is a square pyramid then there as a square is four sides so there will be four sides of the pyramid does that make sense we go with there are differences this list is a little bit .

Larger so uh i'm going to create a little bit of a table so there's going to be a prism side so here it is there you go prism and then we're going to have a pyramid remedy okay so for a prism a prism i'm going to .

Change my uh color we'll do like this so a prism has two bases no matter what the prism will have two bases meanwhile the pyramid will only always have one base okay a pyramid .

Always has triangular faces meanwhile a prism will only have rectangular faces okay okay uh pyramids have slant heights what okay pyramids have slant heights because i mean pyramids are .

Slanted right but a prism does not it just has simple regular height only so it has slant heights that are added to pyramids speaking of height when it comes to the prism the prism you can find the height or the height .

Let me about this is the same as the lateral edge so you can use the lateral edge to determine the height so working your way where did we go can i have like a rectangular prism of some sort this is a hexagon i'll take it so for example .

In this particular hexagonal prism children i know it's a hexagonal prism because the base is a hexagon it has two bases okay so it's not a pyramid and i can find the height by using the lateral edge and this is the this is what you .

Normally see in your geometry sections of your macbook okay and so that's what i'm saying you can use the lateral edge to find the height so that is uh what's happening on the prism and in prisms the lateral edge is the height and then pyramids have slight heights we just .

Said that so these are some differences that you will see between pyramids and prisms so that's important uh i don't want you to call the wrong thing the wrong thing you can yes have hexagonal pyramids it means that the base will be a hexagon .

But all the sides will come to a point and all the sides are triangular faces they're just not very wide okay the more sides you have the more narrow the triangles will be but you can still have triangular prisms or excuse me triangular pyramids so that is what that is please make sure .

That is in your notes so now let's determine whether or not each net will form a pyramid or a uh prism of some sort but what kind if yes we'll name it all right let's see if it makes a pyramid at all uh part a will this make a pyramid yes or no .

Let's see here's the base i'm going to let you know that this is definitely the base there are four sides so it looks like this side goes to here this side if it goes would go here and then this side can form over here so yes it does make a pyramid what kind of pyramid .

A square pyramid they're named for their base and apparently this looks like a square so this is a square i'm gonna be vanna white over here right square pyramid are like those instagram people where they have the product all right peer revenue all right so what .

About part b no no why would it not be anything um yes here's one two three four five six and there's only one two three four five this would not create a correct 3d figure correct that's right what about this would it be and you said yes but why because it has .

The yeah they all same at one point which would be this is the vertex right here one two and three they'll meet at one point when they fold up and there are three sides what's the name of this triangular pyramid yeah because they all come together .

So it's not a prism a prism would have two bases you get the idea so try pyramid so you'll be able to need to recognize next of certain stuff so that we'll throw some nets your way to see whether or not you can see what kind of prism or pyramid it will be .

So let's continue the following just to make sure that we understand the pieces that are involved in the basics here's the vertex the vertex is at point e so everything comes to the point of point e so that's the vertex someone talk to me about the lateral faces .

Now the lateral faces for example would be uh i'm going to use the entire plane so for example one lateral face would be use this whole plane of this right this side right so that would be uh technically plane e b .

C yep see how we're going back to the old school stuff so that's basically a plane all right so then there's ebc what are the other faces e a b yep jesus you're killing it all right and so the e d a and they're also b c e d c .

Would work too edc eda you said i heard y'all kicking butt now c d a b is technically a plane but it's not the lateral face okay so these are the sides so i need those four one two three four i like how e was involved because it's the vertex so it's part of every one of .

Those faces in a way very good now the lateral edges are more specific as line segments see they were going back full circle all the way to the beginning of school where for example line segment d e that is not a d d e .

So what are other lateral edges you know e a yes see how this is easy for you now c e b e yes these are lateral edges this is the basics but that helps us to understand the pieces of what's going on and the base edges are .

Only lining the bottom okay they're only lining the bottom so the base edge could be like a b so what else can be for that b c b c c d a c d d a yeah blazing straight through and those are the base edges of the pyramid .

So let's go to example six we want to find the slant height and here's a skill that we need to be able to do um in this comp course recognizing when we have a pyramid or something especially with pyramids we need to recognize that there's a slice of the pyramid that we'll need to pull .

Out it starts with the height and the slant height and there is a piece that goes from the center to the middle of the side who remembers what is a thing that goes from the center .

To the middle of the side that was called the apothem y'all are kicking butt today i told you it would come back so apothem still exists so we have the apothem that is here i can't write it all of a sudden because it's in the way there it is that okay that didn't work out very well .

There we go we have the apothem and then if i'm labeling something i'm also rambling the fact that this is the slant height okay and the actual height of pyramid is going from the center to the vertex okay and so this would be the height all right so h i 1800 ght all right so cool this is .

The height all right so that is the height so now you have slat height you have the height you have the apothem what we're going to do is pull this out kind of like zoom in a little bit in a way but i want to pull out well pull this this and this how about oh .

I thought i would be able to grab all of it maybe if i did that how about that no come on work with me man all right here we go ah magic there we go all right so what we do know is uh we know the height the height was eight we know the apothem we're not exactly .

Entirely sure but it wait isn't by the definition of the apothem it goes from the center to the middle wouldn't that and if we know this side is 12 it would be half of 12 which is six there you go and so this is 90 degrees this is 90 .

Degrees it's a right triangle oh my gosh what would i do to find the missing side of a right triangle no there's something that i would do to something to equal something something plus something equals something .

That's what we do to find the missing side of a right triangle and we can do that wonderful h squared plus six squared yep will equal the x squared or c squared same and so if you have a calculator let's go ahead and get to calculate so 8 squared is 64. square root .

The 6 squared is 36. that technically equals 100 so 64 plus 36 100 and 100 is equal to x squared so if i were to say 100 is equal to x squared how do i undo something that is squared sorry square root of 4 all righty so so then uh the square root .

Of 100 is one ten there you go what times what is a hundred is ten and so ten is the value of our x which is basically our missing side so now i know what the slant height is so if anybody asked me i now know how to find a slant height .

Of a pyramid okay and this is all this is the fun fun fun part the only thing that was new today was the slant height everything else was stuff we've all done before many many times and now we do it again this is why geometry can be a little difficult .

Because we never learn anything new we technically always just keep doing the same stuff frankly all right and it can help you because you see it again and familiarity is very important so let's try another one if the perimeter of the base .

Now perimeter means that i'm walking along the edge of the figure so perimeter means i'm walking this way then i walk this way then i walk this way then i walk that way and i get to 16 meters the 16 meters is my perimeter what is the value of each side if it's .

A let's assume this is a square oh yeah i know that because it says square based so what is each side going to be there are four sides they'll be divided evenly by four with a sleep side gonna be four so each side is four okay that helps us the base .

Is the perimeter is 16 meters find the height so i need to find from here to there i don't know what that is but i do know that nine is what is nine referring to what is that pointing to is that the lateral height close uh the slant height yes that's the that's .

The word i want you to use the slant height and the lateral height i get what you're saying it's not bad and but i do also remind you that later on you'll see formulas where they use l as the things that might be confusing later on but good job flat height is nine if i were to take this uh triangle and move that slice out .

It would look a little something like this i'm gonna move it this right here it would be at a right angle the slant height we already said was nine what would be this value right here two two because it happens exactly that's the apothem so two and so now we're looking for this oh it's a right triangle .

I'm looking for the missing side of a right triangle i will use the same thing but this time a little differently there's a squared plus b squared equals c squared now if i'm plugging in what i do know i do know that c is 9 so this is 9 squared i do know that let's just say b .

Is 2 or something right that's so that's 2 squared exactly i do the same process in a way but i got to solve for a so really i guess if i took 9 squared and subtracted 2 squared i would get a squared right and so that's what i want you to do .

9 squared which is 81 minus 2 squared which is 4 equals 77 okay and so technically a squared is 77. so if a squared equals 77 what is the value of a if i were to square root both and then round to the nearest one you sure .

So 8.7 something and so if i were to round it it would be something like 9 because i might ask you to round to the nearest whole number or something it's a possibility so we'll see how that works out interesting fact when dealing with the pyramid .

The slant height will always be greater than the real height because it has to travel a farther distance to go from the edge of the base to the middle of the to the middle of the figure so if you ever have a situation where your base height like your real height like this was like .

Let's say that this was 10 that would be wrong let that be a red flag ding ding ding something's wrong this needs to be bigger okay slant height always needs to be bigger than the regular height of the figure so as you do your numbers keep that in mind .

Okay so here we go let's continue on and actually we're done we're kind of done i wanna keep going though but we're gonna do this on another video

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