Geometry Three-Dimensional Figures

Welcome back to Educator.com.0000

For the next lesson, we are going to go over three-dimensional figures.0002

Another word for three-dimensional figures is solids; now, some specific types of solids are polyhedrons.0007

Whenever you have a three-dimensional solid with all flat surfaces, meaning that all of the sides of that solid are flat, then it is a polyhedron.0018

These are examples of polyhedrons: notice how all of the sides are flat--no round surfaces; no circles,0031

because if you have circles as a side, then the other sides that are intersecting with that are not going to be flat.0042

We are going to go into those types of solids more later.0050

But with these, notice that all of the sides are flat, and it has to enclose a single region of space,0054

meaning that it can't be open; it can't be that one side is missing--it has to be closed.0063

The three parts of polyhedrons: the first one are faces; now, the faces are the sides.0069

All of this right here would be a face, and the same thing with all of this, and this is a face...0080

Each one of those is a face--any time you have a side.0095

This one here has four faces: you have 1, 2, 3, and then the one in the back that you can't see; so each side is a face--these are sides.0099

Edges are the segments that are the intersections of the faces.0116

It is where the faces intersect, so these are all edges here; they are segments.0130

And the vertices are all of these; they are the corners, the endpoints of the edges; so these are like corners, the corners of these polyhedrons.0146

Going over solids: the pyramid is the first one: we know that it is a polyhedron, because all of the sides are flat surfaces.0174

So, all of the faces and one base...the base is the bottom part of this pyramid, so if I draw that part,0186

it is considered the base, the bottom part; and each of these sides here is a lateral face.0205

And all of those faces meet at a point, and that is the vertex.0217

We are going to go over each of these more specifically later on.0222

The next is a cylinder; a cylinder is like a can, like a can of soup; it is a cylinder.0226

The bases would be the circles on top and the bottom, so that would also be the base.0236

A cone is one circular base, and a vertex is right here; think of an ice-cream cone.0252

A sphere is like a ball; and this is not called a circle, because circles are two-dimensional; this is a sphere--it is three-dimensional.0265

A basketball would be a sphere; it is a set of points in space that are a given distance from a given point.0279

This point right here is the center of the sphere; and no matter which way you go, no matter which direction, it is always going to be the same distance.0286

The next type of solids is a prism; a prism is when you have two bases that are opposite and congruent, and they are parallel.0300

This is called a rectangular prism; in this case, I can name any two sides bases,0315

because as long as the two opposite sides are parallel and congruent, they would be the bases of a prism.0325

Now, another example of a prism would be like this; now, if you notice, this is not rectangular.0337

So, this would be the base, and this would be the base opposite; these two bases are parallel and congruent, so this is a prism.0366

Now, the base itself is not a rectangle...this would be called a rectangular prism, because the base is rectangular in shape;0382

and for this one, this is a special type of prism because, if I choose this as the base,0398

the side opposite is also going to be parallel and congruent, so these two can be named the base.0409

Or if I want, I can make the top and the bottom the base, or I can make the front and the back the base,0423

because, as long as, in your prism, you have two opposite faces that are parallel and congruent, they can be the bases of your prism.0435

And that would be the type of prism.0446

Two of the faces are bases; the rest of the faces are lateral faces.0451

So in this one, this top and bottom are your bases; the rest of the sides (that means this one right here,0458

this one right here, the one in the back, and then the three other sides) would be your lateral faces.0466

So, bases and lateral faces make up all of the sides, or faces, of the prism.0474

A regular prism is when the bases of your prism...so then, for this one, if I am going to name0483

the top and the bottom the base, they have to be regular.0493

So, as long as the bases are regular (and regular, again, means equilateral and equiangular), so if I am going to call0497

the top and the bottom the bases, if all of these sides are congruent,0516

and all of the angles are congruent, then this would be considered a regular prism.0521

For this one, if this pentagon (because that is the shape of the base) was regular, then this would be considered a regular prism.0531

And if this is regular, then we know that the bottom base is regular, because they are the same.0545

If the base is a regular polygon, then the whole thing, the prism, is a regular prism.0552

And a cube is when each side of a rectangular prism is a square; so that means the front is a square;0559

the top side is a square; the side is a square...if all of these are squares, then this is a cube.0572

Platonic solids: there are five types of platonic solids, and they are all regular polyhedra.0590

"Polyhedra": if we have one, it is a polyhedron; but when we have more than one, it can be either polyhedrons, with an s to make it plural, or polyhedra.0599

So here are the five types: this one right here has 4 sides, the front, this side...there is a back side, and there is the bottom base.0616

So, it will look like this: 1, 2...and then this is the back side that you can't see...0630

So, the one with four sides (and again, this is all regular, meaning that the sides are all regular) is a tetrahedron.0646

The one with 6 (this is a cube, the one with 6 sides) is a hexahedron.0667

This one has 8 regular sides, so this is an octahedron, just like an octagon.0679

This one has 12 sides, so this is a dodecahedron.0693

This one has 20 sides, and that is an icosahedron; those are the five types of regular polyhedra.0707

OK, slices and cross-sections: whenever you have a solid, and you have a plane that intersects the solid, then you have a slice.0728

"Slice" sounds like you are cutting through, so it is intersecting; so here is a cone, and here is a plane intersecting the cone.0743

See how it is cutting through it...so that is a slice.0752

Then, where they intersect, that is the shape that is formed.0759

Now, if you have the plane intersecting the solid so that the plane is parallel to the base0768

(in this case, the plane is parallel to the base; they are both horizontal, so the base is flat like this,0778

and then the plane is flat or horizontal, cutting through somewhere in the middle), then that would be a cross-section.0786

So, when it is parallel, when the parallelogram is slicing so that it is parallel to the base, then it is a cross-section.0798

So in this case, this would be a cross-section.0808

And the figure, or the shape, that is formed by the intersection would be a circle.0811

Now, you can see here that it is like this; that is where they are intersecting.0817

But that is because it is kind of sideways; if you were to draw it from a top view,0823

meaning if you had a counter right here and you are looking down this way, then you would see a perfect, round circle.0827

That is not perfect, but it is a circle; so that is the shape that resulted from the slice, from the intersection of the plane and the cone.0835

So then, again, when the plane intersects the solid, it is slicing it, and the shape that resulted from that is a circle.0848

Moving on to our examples: name the edges, bases, and vertices of the polyhedron.0865

Remember: edges are all of the sides, all of the segments where all of the faces intersect.0873

So, for the edges, I have to just name all of these segments; so I have AB; I have AE; let me just do that,0884

so that I know that I wrote it down; AE, and then there is that one, BE, this one, AF, BC, ED0905

(it doesn't matter if you do ED or DE), CD, CF, and FD; so then, I have 1, 2, 3, 4, 5, 6, 7, 8, 9--those are all of my edges.0928

And then, bases would be triangle ABE and then the triangle CFD.0959

Then, I am going to name this side here (that is ABCF), then this one here, which is AFDE, and (which one?) this one right here on this side; that is BEDC.0985

Make sure that, when you label it, you label it in order: BEDC--it can't be BDCE--it can't go out of order (but you can do BCDE).1014

And then, we have vertices: vertices would be just these corners, so it would be vertex A, vertex E, vertex B, vertex F, vertex C, and D.1029

Those are the edges, faces, and vertices.1053

The next example: Determine if the figure is a polyhedron, and explain why.1058

This is a cone; it has one base, which is a circle, and we have a vertex.1064

Now, this is a solid, because it is three-dimensional; but it is not a polyhedron, so this would be "no," because polyhedrons have only flat surfaces.1074

Now, this base is flat; it is a flat circle; but how about the cone--isn't it circular? It is not flat.1094

There is no side to this, so it is "no," because it is not a flat surface.1105

So, the solid, because there are no faces, does not have flat surfaces as the sides.1116

Even though, again, the bottom one, the base, is flat, the side here is not flat; it is going circularly.1140

OK, the next example: Describe the slice resulting from the cut.1154

As you can see here, this cone is being cut, or sliced, sideways, like this; so the parallelogram is cutting into the side.1159

Now, what is the shape that has resulted from this cut--what is that cut going to look like, where they intersect?1172

Well, if you have a cone here, it is going to go like this; it is cutting right there--that is a part of it;1184

and then, what about the other side?--it is probably going to go like that, something like this.1197

So, this shape would be a parabola; it is going to be like a U--something like that; that is the shape that is from the slice,1207

from the intersection of the plane and the cone.1224

Just another example: if you have, let's say, a cylinder, and let's say you have a plane that is intersecting,1232

or slicing, that cylinder; well, what is the shape that resulted from the slice?1246

Again, if you look at it from a top view, it is going to be a circle.1261

Think of a can, and you cut in half down the middle; you get a knife, and you just cut it through the middle; aren't you going to get a perfect circle?1272

So, that is another example of a slice.1280

Describe the shape of the intersection: A cube is intersected by a plane that is parallel to the base.1288

A cube is being intersected so that it is parallel to the base; so then, my plane will be something like that.1298

It is being intersected right here; and it is parallel to the base, so that would be a perfect square.1328

A sphere is intersected by a plane at the center of the sphere.1349

I have a sphere; I don't know how you would show a sphere; and it is being intersected right here by a plane.1355

As a result, it is going to look like this; and it has to cut through the center, so what is the result from this slice?1381

It would be a perfect circle, like that.1395

Again, if you look at it from your screen, you are going to see that it is going to look like this; but you have to look at it from a top view.1401

And it is going to be a perfect circle.1412

And that is it; thank you for watching Educator.com.1416