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### Surface Area of Prisms and Cylinders

- Prisms: Two bases formed by congruent polygons that lie in congruent planes
- Lateral faces are not bases and are formed by parallelograms
- Lateral edges are formed by the intersection of two adjacent lateral faces
- The altitude measures the height of the prism
- Right prism: A prism whose lateral edges are also altitudes
- Oblique prism: A prism that is not right
- Lateral area: The area of all the lateral faces of a prism
- Surface area of a prism: The sum of the areas of its outer surfaces, or lateral area + 2(area of the base)
- Cylinders have two bases that are parallel and congruent circles
- The axis is the segment whose endpoints are centers of the circle
- An altitude is a segment that measures the height
- In a right cylinder, the axis is the same as the altitude; in an oblique cylinder, the axis is not the same as the altitude
- Lateral area of a right cylinder = 2πrh
- Surface area of a right cylinder = lateral area + 2(area of the circle)

### Surface Area of Prisms and Cylinders

All the lateral faces of a polyhedron are parallelograms.

All the lateral edges are the altitude of a prism.

Name the prism, ABCD and EFGH are rectangles and they are congruent to each other, and ―AE , ―BF , ―CG , ―ED are altitudes.

Find the lateral area of the right rectangular prism, ABCD is a rectangle.

- LA = perimeter of base * height
- LA = (4 + 4 + 6 + 6) * 7

Find the surface area of the right rectangular prism, ABCD is a square.

- LA = perimeter of base*height
- LA = (6*4)*7 = 168
- SA = LA + 2(area of base)
- SA = 168 + 2*(6*6) = 240

determine whether the following statement is true or false.

If all the edges of this prisim are congruent, then it is a cube.

Find the lateral area of the right cylinder.

- LA = perimeter of base * height
- LA = (2πr)*h
- LA = (2*3.14*4)*10

Find the surface area of the right cylinder.

- LA = perimeter of base * height = (2π r) * h
- LA = (2*3.14*5)*12 = 376.8
- SA = LA + 2(area of base)
- SA = 376.8 + 2*(3.14*5*5) = 533.8

Find the lateral area of the right prism.

- LA = perimeter of base * height
- LA = (4 + 4 + 8 + 6 + 16 + 6)*20

find the surface area of the right prism, ―AB ⊥―BC , AFED is a rectangle.

- AC = √2 AB = √2 *4 = 5.66
- AD = AC + CD = 5.66 + 8 = 13.66
- EF = AD = 13.66
- LA = perimeter of base * height
- LA = (4 + 4 + 8 + 6 + 13.66 + 6)*20 = 833.2
- Area of a base = area of ∆ABC + area of AFED
- Area of a base = [1/2]*4*4 + 6*13.66 = 89.96
- SA = LA + 2(area of a base) = 833.2 + 2*89.96 = 1013.12

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Surface Area of Prisms and Cylinders

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Prisms
- Prisms
- Classifying Prisms
- Lateral Area of a Prism
- Surface Area of a Prism
- Cylinder
- Lateral Area of a Cylinder
- Surface Area of a Cylinder
- Extra Example 1: Find the Lateral Area and Surface Are of the Prism
- Extra Example 2: Find the Lateral Area of the Prism
- Extra Example 3: Find the Surface Area of the Prism
- Extra Example 4: Find the Lateral Area and Surface Area of the Cylinder

- Intro 0:00
- Prisms 0:06
- Bases
- Lateral Faces
- Lateral Edges
- Altitude
- Prisms 2:24
- Right Prism
- Oblique Prism
- Classifying Prisms 3:27
- Right Rectangular Prism
- Oblique Pentagonal Prism
- Right Hexagonal Prism
- Lateral Area of a Prism 7:42
- Lateral Area of a Prism
- Surface Area of a Prism 13:44
- Surface Area of a Prism
- Cylinder 16:18
- Cylinder: Right and Oblique
- Lateral Area of a Cylinder 18:02
- Lateral Area of a Cylinder
- Surface Area of a Cylinder 20:54
- Surface Area of a Cylinder
- Extra Example 1: Find the Lateral Area and Surface Are of the Prism 21:51
- Extra Example 2: Find the Lateral Area of the Prism 28:15
- Extra Example 3: Find the Surface Area of the Prism 31:57
- Extra Example 4: Find the Lateral Area and Surface Area of the Cylinder 34:17

### Geometry Online Course

### Transcription: 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 sides*0754

*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 be*0958

*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 πr ^{2}, so it is 2 times πr^{2}.*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, h ^{2} + 2.5^{2} = 6^{2}; 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 πr ^{2}; 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

0 answers

Post by Shahram Ahmadi N. Emran on July 10, 2013

Thanks