Geometry Surface Area of Prisms and Cylinders

Welcome back to Educator.com.0000

For the next lesson, we are going to go over the surface area of prisms and cylinders.0002

First, let's talk about prisms: prisms, we know, are solids with two bases that are both parallel and congruent.0008

They are congruent polygons that lie in congruent planes; they have to be parallel and congruent.0025

If you look at this prism here, this top and this bottom right here would be the bases; they are both parallel and congruent.0038

Now, the other sides, the remaining faces, are lateral faces; they are all of the faces that are not the bases.0054

And each of them, each of those lateral faces, would be a parallelogram.0064

So, if you notice, this is a parallelogram, parallelogram, parallelogram; all of them will be parallelograms.0070

Lateral edges are those line segments where the faces intersect; so this face and this face intersect here--those are lateral edges.0080

And the lateral edges would be only from all of the lateral sides, where the lateral sides intersect--not these.0096

These are considered edges, too, but they are not lateral edges, because these are edges from the base.0106

It would be all of these here; those are lateral edges.0113

And the altitude measures the height of the prism; so altitude, we know, has to be perpendicular to the segments.0120

This perpendicular segment is the altitude; it measures the height, because height, we know, has to be perpendicular.0131

The two types of prisms: it is going to be either right or oblique.0146

A right prism is when these lateral edges are altitudes; so the measure of the lateral edge, the length, is the height of the prism.0151

So, this right here measures the height; then it is considered a right prism, because it is standing up (right).0167

An oblique prism is one that is tilted; it is slanted to the side.0177

So, in this case, these lateral edges are not perpendicular; they are not considered altitudes, so they are oblique; that is a prism that is not right.0181

So then, if you have to find the height of this, then you would have to find the height that is perpendicular to the base.0195

Classifying prisms: classify a prism by the shape of its bases.0209

Depending on the bases, there are different names for these prisms.0216

These are all prisms; all four of these are just a few types of prisms.0220

The first one: if we label the top and the bottom as bases...0227

now, for this rectangular prism, it is a special type of prism, because we can actually name any pair of opposite sides as its bases,0235

because we know that bases just have to be congruent and parallel;0245

and for this, each pair of opposite sides is congruent and parallel.0251

So, for a rectangular prism, it doesn't matter which two opposite sides you label as bases.0256

But if it is standing this way, then just to make it easier, you can name the top and the bottom as bases.0263

Then, this would be a right (because it is standing upright; all of the lateral edges are perpendicular) rectangular prism.0270

This one here...now, be careful here, because in this case, the bottom is not considered the base.0297

This triangle right here, with this front side and this back side, would be considered the bases.0310

The other sides, the bottom, the right side, and the left side, are all lateral faces.0327

So, don't always think that the top and the bottom are going to be the bases; in this case, it is the front and the back, and they are triangles.0336

The bases are triangles; that would make this...0345

and for determining whether it is right or oblique, if we were to take this solid and stand it up,0349

so that the bases were the top and the bottom sides, then this would be the height; it would be the altitude.0357

And it would be perpendicular to the bases; so this is also a right...0367

and then, the shape of the base is a triangle; so this is a triangular prism.0374

This one here, we know, is oblique, because we can just tell that it is not standing up straight; it is slanting to the side, so this is oblique.0388

And then, the bases would be this top and the bottom; and they are 1, 2, 3, 4, 5...5 sides, so that is a pentagon: pentagonal...this is "al"...prism.0404

And this last one here, we know, is right; and how many sides is the base?0435

We know that this top and the bottom is a base, again; 1, 2, 3, 4, 5, 6...so that is a hexagon, so that is a hexagonal prism.0443

OK, to find the lateral area of a prism, first make sure that you determine which sides your bases are and what your lateral faces are.0464

And once you do that, your lateral area is just the area of the lateral faces.0478

If we say that the top and the bottom are the bases, that means that we are finding the area of all of these four sides.0487

Left, front, right, and back--we are looking at the area of all the four sides, minus the bases--not including the bases--and that is lateral area.0498

The formula to find the lateral area of a right prism would be the perimeter of the base, times the height.0516

And the reason for this formula is (I am going to explain it to you): let's say that we take scissors, and we cut one of these sides.0524

Let's say you cut it there--cut that corner--and you unfold it.0545

When you unfold it (and I have a paper here to demonstrate), here is the rectangular prism.0551

We know that it is a rectangular prism, because the bases are a rectangle.0565

If you were to cut it, and you unfold, you get a rectangle; so again, this is a rectangular prism;0572

if you cut it and unfold, then you get a rectangle.0592

This is my cut that I made; you end up getting just a big rectangle.0605

So, if this is side 1, this right side; this front side is side 2; this is side 3 on the left; and the back is side 4;0615

well, it is as if I have side 1, side 2, side 3, and side 4; and actually, this cut is made to this side, the left side, or the right side of side 1.0627

So, it is as if this would be side 1; so either way, it is just a big rectangle.0651

So, each of these sides, 1, 2, 3, 4, 1, 2, 3, 4...then, when you fold it back up, it will be like this, with side 2 folding this way, this way, and this way.0659

It is as if you are taking this, and you are folding it back over.0671

Now, in that case, since the lateral faces all make up the big rectangle (I am going to erase this, so you don't get confused),0676

let's say that this right here has a measure of 2, and let's say that each of these is the same;0702

well, then I know that this is 2; this is 2; 2; and 2; and the height, the altitude, is, let's say, 10.0711

Then, I know that this right here would be 10.0724

The perimeter here: to find the area of this, it would be all of this length right here, times this.0730

Here, this is 2, 4, 6, 8; this has a measure of 8, and this is 10; so the area of all of these lateral faces would be 80.0741

So then, that is how this formula came about, because, if you were to cut it, well, then the area of all of the lateral sides0754

would be the perimeter of the base (because you are doing this, this, this, and this) times the height,0766

because if you were to unfold it, this, that, and all of these sides would come up to 8.0772

And isn't that the perimeter of the base?0779

So, that is why it is the perimeter of the base, which is this, times the height; and that is 10.0787

So then, the lateral area of this would be 80 units squared, because it is still area; you are finding all of the space.0803

That is 80 units squared, and that would be only the lateral area.0814

Now, next, let's go over surface area: the surface area is the area of all of the sides, so it would be the lateral area, plus the area of the two bases.0824

It is the lateral area, plus 2 times the area of the base.0845

This top right here is the base, and then this right here is also a base.0851

Now, if you want, you could find the area of each one of these: 1, 2...all of the sides, all of the bases, and then just add them all up.0858

That would give you the surface area, because it is the sum of the areas of its outer surfaces--all of the sides.0869

But since we know that the lateral area would be the perimeter--so again, if you were to cut it,0877

let's say, right down here, and unfold the lateral area, then it would just be the perimeter,0885

because it would be this side, this side, this side...all of those sides, which is the perimeter, times the height.0895

It is going to just give you one big rectangle; the perimeter of the base is going to be the length; so this is the perimeter of the base,0911

and this is the height; so that, plus...and then the area of the base, times 2, because you have 2 bases.0933

And then, that would give you the area of all of the sides together.0951

That is surface area: lateral area is just the area of all of the lateral sides, and then the surface area would be0958

the area of all of the outer sides, including the bases; that is lateral area, plus the area of the bases.0967

Next, we have cylinder: a cylinder has two bases that are parallel and congruent circles.0979

So, it is like the prism, except that the bases are circles instead.0986

Here is the base here, and here.0990

The axis is the segment whose endpoints are the centers of the circles.0994

So, it has to go from the center of one base to the center of the other base; that is the axis.1003

Now, it could be different from altitude; now, in this case, in a right cylinder, the axis is the same as the altitude, because the altitude measures the height.1013

If it is standing up straight, then it doesn't matter where the endpoints are--center to center or from end to end.1024

As long as the endpoints are on the two circles, that is the altitude.1037

For a right cylinder, altitude is the same thing as the axis; in an oblique cylinder, that is not the case.1042

The altitude is right here; that measures the height; this is the altitude.1056

But the axis, remember, has to go from the center to the center of the two circles; so this right here is the axis.1064

It is not the same in an oblique cylinder.1077

Now, the lateral area of a cylinder is the same concept as the lateral area of a prism; lateral area is the same.1084

To find it within the cylinder, again, it is just the area of everything but the bases.1094

So, we know that this is the base; we know that this is the base; so it would be the area of just the outer part.1103

Now, think of a can, like a soup can; and you tear off the label--the label goes around the can.1113

It is like finding the area of that label; that would be like lateral area.1121

Again, if you make a cut, like this paper here--if you have a cut--there is your cylinder without any bases,1129

and you cut it, and then you open it up, you are going to get a rectangle.1147

To find the area of the rectangle (the lateral area just means that you are finding the area of a rectangle),1163

you need base, and you need height; the base would be (if I turn it back into a cylinder)...1169

isn't the base...let me actually call it the width--the width and the length--so that you don't get it confused with these bases.1182

The width of the rectangle is the same as the measure of this other circle.1198

What is that called? That is called circumference: to measure this all right here, that is the circumference.1206

The width of this rectangle is 2πr, because it is the circumference--just 2πr.1218

And then, the length, we know, is h; it is the height of that.1233

That is how we get this formula here: 2πr times the height--that is the lateral area.1241

And for surface area of a cylinder, the same thing works: lateral area, plus the area of the two bases.1255

Now, this is the easiest way to remember it, because the lateral area is always just going to be a rectangle.1268

And then, we just find the area of the circle, and the area of the other circle--1277

or find one of them, and then, since they are going to be the same, just multiply it by 2,1285

and then add them together: so it is this area, times 2, plus that; all of these together is going to equal the surface area of the cylinder.1290

It is going to be πr², so it is 2 times πr².1303

Let's do some examples: Find the lateral area and surface area of the prism.1312

Now, the first thing to do is to figure out what the bases are.1319

Whenever you have a prism, the easiest way to point out the bases is to look for any shapes, any sides or bases, that are not rectangular.1329

Automatically, we know that these two triangles will be the bases.1342

Now, if you have a face that is not rectangular, but there is only one of them, then it can't be a prism; it is going to be something else.1351

It will probably be a pyramid or...I don't know; it is not going to be a prism.1360

If it has two opposite sides that are congruent and that are not rectangular, then those two sides would be the bases of the prism.1369

Those are my bases; and then, the lateral area--imagine the cut; you are going to unfold; it is going to be one rectangle.1384

And that is three sides, three lateral faces, that make up the rectangle, this lateral area; the three sides go like that.1396

One side is 6--that is one side; the other side is 6, because this side and this side are the same; and then, the other side will be 5.1412

So, if this is where the cut is made, then it is as if this is where the cut is made; so 6 + 6 is 12, plus 5 is 17;1434

so this whole thing right here is 17, and then the height is 8.1461

We are going to do 17 times 8 to find the lateral area: on your calculator, do 17 times 8, which is going to be 136 units squared.1470

That is lateral area; then, surface area--all I have to do for surface area, since I have my lateral area,1496

is to find the area of the base, multiply it by 2 (since I have 2 of them), and then add it to this lateral area.1501

Find the area of the triangle: now, to find the area of this triangle, it is going to be 1/2 base times height; this is 6; this is 6; and this is 5.1510

I need the height; now, for the height, because this is an isosceles triangle, I know that this is half of this whole thing,1525

so this will be 2.5, half of 5; then, to find the height, to find h (this is 2.5, and this is 6), I can just use the Pythagorean theorem.1539

So, h² + 2.5² = 6²; 2.5 squared...h squared is 29.75;1554

take the square root of that, and my height is 5.45.1580

1/2...my base is 5, and then, for this base, make sure that it is the whole thing; it is not just this half.1596

We only use that half just to look for the height; we can use the Pythagorean theorem for the height and make this a right triangle.1611

But when it comes to the actual triangle, we are finding the area of the whole thing, so you have to use 5 as the base.1618

And the height is 5.45; I am just using my calculator that I have here on my screen; I get that my area of this triangle is 13.64.1624

My surface area is going to be 136 + 2 times the area of the triangle, and I got 13.64.1652

Multiply it by 2, and add it to 136; and I get 163.27 units squared.1668

That is the lateral area and surface area of this prism here.1686

The next example: Find the lateral area of the prism.1697

Now, this doesn't look like a prism; it looks kind of odd-shaped.1700

But remember: as long as you have two sides that are opposite and congruent, it doesn't matter what shape it is; those are the bases of the prism.1705

In this case, for this solid, we have the front and the back as the bases; this whole thing right here is considered the base.1719

So, that way, each of the lateral faces is rectangular; they are all rectangles.1732

Again, if you were to take this solid and stand it up, so that the bases would be the top and the bottom,1743

and then make a cut like this, it is going to be the perimeter...if you unfold it, it is one long rectangle;1753

and then, this right here, the length of the lateral area, is going to be the perimeter of the base.1769

So, it is going to be like this--all of this is going to make up this whole thing right here.1778

There is our cut; then this is like having a 1; and this is 5; that is this and then this, and then that would be the 4; and so on, all the way through.1785

I am going to do perimeter as 1 + 5 + the 4 + 8 +...and then, what is this side here?1809

This side would be the 4 plus the 1; this is 5, plus...and then, this whole thing right here is the 8, plus the 5; that would make this...and that is 13.1825

The perimeter I get: this is 6, plus 4 is 10, plus 8 is 18, plus 5 is 23, plus 13 is 36.1843

So, this is 6; this is 10; this is 18; this is 23; and then, together, they are 36.1859

And so, after that, we need to find the height; this whole thing right here is 36, and then what is this height right here?1869

That is 2, because all of these have to be the same; so my lateral area is going to be 36, my length, times the width, and that is 2;1881

so, that will be 72 units--whether it is inches, feet, and so on--squared.1900

And then, let's find the surface area of that same figure.1916

The lateral area was 72 units squared; then I want to find the area of my base, because, remember:1925

surface area would be the sum of all of the sides, so it will be lateral area, plus the area of the base, plus the area of the other base.1937

So, it is 2 bases; we have to add both to the lateral area to get surface area.1949

Here, to find the area of this base right here (just this front--this is a base), I need to break this up,1955

because there is no way that I can find the area of that, unless I break it up into 2 polygons, like that.1965

Here, this will be 8 times...what is that?..this is 4, and then this is 1, so then this would be 5.1975

So again, length times width here--this is 40 units squared, and then, for this right here, it would be 5 times 1; so that is 5 units squared.1987

Then, I add these together, and this would be 45; so the area of the base is 45 units squared.2006

But then, since I have two of them--I have a front, and I have a back--my surface area is going to be my lateral area,2020

all of that, plus two times my base; so that is 45; so 72 +...2(45) is 90...2028

and then this will be 162 units squared; this is my surface area, then.2044

The fourth example: Find the lateral area and the surface area of the cylinder.2060

Now, in the same way as our prism, if we make a cut right here, and lay it out flat, then it will just be a rectangle,2065

whereas this is the circumference, because this measures from here all the way around here, and that is the circumference.2086

So, it is 2πr, and then this is the height, which is 9.2098

2πr is 2 times π times...the radius is...4; that is the length; the width is 9.2107

Multiply that together; 2 times 4 times 9...you get 226.19...now, I am just rounding to the nearest hundredth, 2 places after the decimal.2126

That would be the lateral area; now, you can probably just leave it in terms of π, if you can.2169

2 times 4 times 9: 2 times 4 is 8, times 9 is 72; so you can probably leave it as 72π units squared.2179

But otherwise, if you have to solve it out, then you can just use your calculator: 72 times π, which is 3.14.2193

This would be the answer for the lateral area.2201

And then, to find surface area, we are going to find the area of the base, which is a circle.2205

The area of a circle is πr²; π times r, the radius, is 4 squared, which is π, or 3.14, times 16.2213

16 times π is 50.27 units squared, and that is the area of one of the circles.2237

But since I have two of them, I need to multiply this by 2; so my surface area is my lateral area, plus 2 times the area of the base.2255

And I am going to put a capital B there, to represent the area of the base:2272

this is 226 (I am going to use this number up here) and 19 hundredths, plus 2 times 50.27.2277

And then, using your calculator, solve that out; and you should get 326.72 in...don't forget...units squared; that is the surface area.2292

OK, well, that is it for this lesson; thank you for watching Educator.com.2327