For more information, please see full course syllabus of Basic Math

For more information, please see full course syllabus of Basic Math

## Discussion

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### Surface Area of a Cylinder

#### Related Links

- Surface area of a cylinder: The sum of the lateral area and the two bases
- Surface area = Lateral area + 2(Area of the circle)

### Surface Area of a Cylinder

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 0:00
- Surface Area of a Cylinder 0:06
- Introduction to Surface Area of a Cylinder
- Surface Area of a Cylinder 1:33
- Formula
- Extra Example 1: Find the Surface Area of the Cylinder 5:51
- Extra Example 2: Find the Surface Area of the Cylinder 13:51
- Extra Example 3: Find the Surface Area of the Cylinder 20:57

### Basic Math Online Course

### Transcription: Surface Area of a Cylinder

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over the surface area of a cylinder.*0002

*Remember surface area is the area of all the sides of the solid.*0007

*For the cylinder, we are going to find the area of the top base, the bottom base,*0013

*and then the lateral area which is the rectangle which makes up the side of the cylinder.*0023

*Imagine if this is a can of soup.*0035

*If you were to rip off the label, the label that wraps around the can, this is a cylinder.*0041

*If you were to cut it and rip off the label, it becomes a rectangle.*0053

*Here the lateral area, meaning this side that does not include the bases,*0061

*if you cut it and open it up, then it becomes a rectangle.*0071

*The lateral area for the cylinder is the rectangle that makes up the side of the cylinder.*0081

*Then we have of course the two bases which are circles.*0089

*Surface area is the lateral area plus the two bases, the two circles.*0095

*To find the lateral area, if this is where we cut, then there is the cut right there.*0105

*We need to find the measure of this length and width*0121

*to be able to find the area of this lateral area, this rectangle.*0126

*Here we know that this wraps around this base.*0131

*All of this here becomes this here.*0137

*When you are given this cylinder, to find this measure, to find that length,*0145

*we have to be able to find the length of this circle right here.*0151

*If we had a prism, that would be considered the perimeter of the base.*0158

*But because it is a circle, that is called circumference.*0162

*To find the circumference of a circle, it is going to be 2πr.*0167

*This is 2πr; circumference, all of this; circumference equals 2πr.*0175

*π is 3.14; there is the r; there is the radius.*0188

*Once you find that, that will be the length of this right there.*0195

*We know this would be the height.*0199

*That would just be the height of the cylinder.*0202

*To find the area of a circle, because we have to find the area of this whole thing.*0206

*This whole thing is this whole thing right here.*0210

*The area of a circle is πr ^{2}; a few things to look at here.*0215

*We have circumference to help us find the length of this rectangle, the lateral area.*0223

*We have the area of a circle because that is the area of the base.*0231

*Be careful here not to get confused with volume.*0236

*Surface area and volume are very different things.*0239

*Surface area, we are just finding the space of the outside of the cylinder.*0242

*Whereas volume measures everything inside, how much space it is covering on the inside.*0249

*Surface area will be the lateral area; lateral area is this times this.*0257

*It is circumference times the height; or I can just say 2πr times h.*0270

*That is going to give me the lateral area, all of this here.*0282

*Plus... now I have to find the area of my bases.*0286

*Plus, since I have two bases, 2 times πr ^{2}.*0291

*The surface area SA is going to be all this here.*0301

*Try not to let this confuse you; surface area is just very simple.*0306

*You are just finding all the space on the outside.*0311

*Just this right here, the area of this rectangle, plus the area of the circle which is the top.*0319

*Plus the area of this bottom circle which is the bottom.*0329

*It is just area of the rectangle plus the two bases of the cylinder.*0333

*That is the surface area.*0339

*As long as you are comfortable, you know the circumference of a circle,*0341

*you know how to find the area of a circle, you are fine with surface area.*0345

*Let's do a few examples.*0351

*Again try not to get confused with the formula for volume.*0354

*The volume is the area of a circle πr ^{2} times the height of the cylinder.*0359

*That is the volume.*0370

*For surface area, we are just going to take it one step at a time*0371

*and find the lateral area, find the area of that label that wraps around the can.*0377

*Then add it to our bases, the area of the bases.*0383

*Surface area, we are going to do lateral area which is that label*0388

*plus the area of the base which is a circle, πr ^{2} since it is a circle,*0394

*and multiply that by 2 because we have two of those bases.*0401

*Again remember a lateral area, we are finding the rectangle; that label, rectangle.*0407

*If we cut it here, then it is as if we are cutting it here.*0416

*We need to find this measure here which would be this measure there which is the circumference.*0425

*The circumference is 2πr; that is 2 times 3.14 times... the radius is 4.*0433

*2 times 3.14 is... 2 times 1 is 2; this is 6.*0453

*I have two numbers behind decimal points; that will be 6.28.*0467

*Times 4; this is 32; that is 8, 9, 10, 11; 24, 25.*0471

*Again two numbers; it is 25.12.*0484

*Easier way, you can just multiply this number with this number first.*0488

*That is 8; then you can multiply that 8 to 3.14.*0494

*Here our circumference is 25.12; this is 25.12.*0500

*That is the length of this side right here; this height is 6.*0506

*The lateral area... I am going to write that in here.*0516

*Lateral area is 25.12 times 6; lateral area.*0521

*25.12 times the 6; this is 12.*0542

*6 times 1 is 6; plus 1 is 7.*0551

*5 times 6 is 30; 6 times 2 is 12; plus 3 is 15.*0554

*How many numbers do I have behind decimal points?--I have two.*0566

*That means I have to place two numbers behind the decimal point here in my answer.*0570

*25.12 times 6 is 150.72; that is only this part right here.*0576

*Plus... now I have to find the area of my base, πr ^{2}, all of this right here.*0589

*πr ^{2}; the area of the base is πr^{2}.*0603

*That is 3.14 times the radius which is 4 ^{2}.*0613

*Remember that I have to square this first; 4 times 4 is 16.*0620

*3.14 times 16; 4 times 6 is 24.*0627

*6 times 1 is 6; plus 2 is 8; 6 times 3 is 18.*0640

*Leave this space alone; 1 times 4 is 4; this is 1; this is 3.*0647

*I am going to add them up; 4, 12, 10, 5.*0653

*I have a total of two spaces behind decimal points.*0661

*I have to put that decimal point right there; it becomes 50.24.*0666

*The area of this base right here πr ^{2} is 50.24.*0671

*I just place that right here, that number, 50.24.*0684

*Since I have two of them, I have to multiply that by 2.*0692

*If you want, you can just do 50.24 plus 50.24; times 2.*0699

*4 times 2 is 8; this is 4.0; that is 10.*0712

*All this lateral area plus 100.48 is surface area; add these two numbers together.*0722

*Remember when you add decimals, you have to line up the decimal point.*0742

*2 plus 8 is 10; 7 plus 4 is 11; plus the 1 is 12.*0749

*1, 5, and 2; decimal point, just bring it straight down.*0760

*My surface area is 251.20; my units, centimeters; it is area so it is squared.*0766

*I know this was a lot of work; this was a lot.*0780

*But all we did is find the area of that label.*0783

*Remember that rectangle that wraps around the can.*0787

*We are going to find the area of that; it is just a rectangle.*0791

*You are just finding 2πr because that is this side right here; times the 6.*0794

*If you want, instead of using this whole formula here, you can just maybe draw out the rectangle.*0802

*Draw your two bases, your circles; find the area of each one.*0810

*Then just add them together; that is all we did here.*0816

*This is just rectangle plus the base times 2; that is surface area.*0819

*Let's do a couple more examples; find the surface area of the cylinder.*0829

*For this one, let's draw out our net.*0836

*Meaning I am going to draw out my rectangle which is that label that wraps around the can.*0843

*Then my two circles.*0851

*First let's find the area of that rectangle; this I know is 20 meters.*0859

*Then to find this measure here, that is just from here going all the way around.*0871

*That is remember circumference; this is going to be 2πr.*0879

*2; π is 3.14; r, my radius is not given; I have my diameter instead.*0889

*The diameter is remember from here all the way to here is 12.*0898

*My radius is half of that; r is 6; it is times 6.*0903

*To find this, I can just multiply these two first.*0912

*2 times 6 is 12; 12 times 3.14; 3.14 times 12.*0918

*This is 8, 2, 6; leave the space alone; 1 times 4 is 4.*0930

*That is 1; that is 3; now we add; 8, 6, 7, 3.*0940

*I have two numbers total behind decimal points.*0950

*Place the decimal point in front of two numbers.*0953

*This side right here, my circumference, is 37.68.*0957

*To find the area of this, just multiply this number times that number.*0966

*It is 20 times 37.68; 37.68 times 20.*0971

*If you want, you can drop the 0 and just multiply this number by the 2.*0986

*But you have to remember that in your answer, after you multiply it by 2,*0994

*you have to place a 0 at the end of that.*0997

*That is just a little faster way to do it.*1000

*Or I can just say that this is 0.*1004

*Then instead of placing... because everything times this is all 0.*1008

*Instead of writing all these 0s, I can just go ahead and start multiplying this next number, the 2.*1011

*2 times 8 is 16; this is 12, 13, 14, 15; 6, 7.*1018

*Two numbers behind decimal points; right there.*1031

*The area of this is 753.6 meters squared.*1037

*That is the area of this rectangle there.*1054

*Now let's find the area of the base, this right here, the circle.*1058

*That will be πr ^{2}; π is 3.14; radius is 6^{2}.*1065

*Again exponent first; that is 3.14 times 36.*1080

*4 times 6 is 24; 6 times 1 is 6; plus 2.*1096

*6 times 3 is 18; put a 0 there; 3 times 4 is 12.*1102

*3 times 1 is 3; plus the 1; 4; 9.*1113

*Add them; 4; 10; 12, 13; 9, 10, 11; two numbers there.*1120

*My area of this circle is 113.04 meters squared; that is the area of that.*1134

*The area of this is also 113.04.*1150

*Now all I have to do to find the surface area is add them together.*1158

*753.6 or 60... remember if this is a 0 that is behind the decimal point*1163

*and at the end of a number, you can just drop it.*1175

*Plus my circles; 113.04 plus 113.04.*1180

*Let's go ahead and just add them; line them up.*1196

*753.60 with this one, 113.04 and 113.04.*1199

*Make sure when you add decimals, the decimal points all line up.*1213

*The decimal point in the answer is just going to go straight down.*1218

*This is 8; 6 plus 0 plus 0; 6.*1222

*3, 6, 9; 5, 6, 7; 7, 8, 9.*1228

*Our surface area is 979.68 meters squared for area.*1238

*Third example, we have a cylinder that has a radius of 6 inches and a height of 10 inches.*1260

*Draw a net for the cylinder and find the surface area.*1267

*Cylinder, there is my cylinder there.*1274

*My radius is 6 inches; my height is 10 inches.*1283

*The net is going to be rectangle for the lateral area and then my two circles, my bases.*1294

*The surface area is going to be this plus this plus that.*1310

*Remember to find the length here, that is going to be*1319

*from here all the way wrapping around which is the circumference.*1326

*That is 2πr; the radius is 6.*1333

*This is going to be 2 times 3.14 times 6.*1339

*I can go ahead and multiply this 2 and the 6 together.*1347

*That is 12; times the 3.14; that is going to give this length right there.*1350

*3.14 times 12; 8; 2; 6; 1 times 4.*1359

*Leave this space alone; put a 0 there; 4, 1, and 3.*1369

*Add them; it is 8; this is 6; 6, 7, 3.*1376

*I have two numbers behind decimal points here.*1385

*I am going to place the decimal point right there in front of two numbers.*1388

*My circumference here, this length is going to be 37.68; my height is 10.*1397

*To find the area of this rectangle, I have to multiply this times this, length times the width.*1409

*Here if you just multiply by 10, all I have to do is*1418

*take the decimal point and move it over to the right one space.*1424

*The area of this rectangle is going to be 376.8.*1429

*It went from in front of the 6 to behind the 6; one space.*1436

*Then plus... I have to find the area of that circle.*1443

*The area of a circle is πr ^{2}.*1448

*π is 3.14; the radius is 6; 3.14 times 6 ^{2}.*1453

*Remember we have to take care of this exponent first.*1465

*6 times 6 is 36; it is 3.14 times 36.*1468

*This is 24; this is 6; add the 2; 8; this is 18.*1479

*Leave the space alone; put a 0.*1488

*3 times 4 is 12; 3 times 1 is 3; plus 1 is 4.*1490

*3 times 3 is 9; 4; now we are adding.*1496

*8 plus 2 is 10; this is 12, 13; 9, 10, 11.*1503

*Two numbers behind decimal points; it is going to go right there.*1512

*The area of this circle is 113.04.*1517

*This is the same circle, same area; this is 113.04 also.*1526

*To find the surface area, I take this which is called my lateral area.*1533

*Lateral area plus area of my base; then add another base because I have two of them.*1540

*If you want, you can just take this and multiply it by 2 since we have two of them.*1553

*I just have to add this all up.*1558

*Remember when we add decimals together, we have to line up the decimal point.*1563

*Decimal point has to go right there; that is 04; again 113.04.*1569

*I am missing a number right here; I can just place a 0.*1581

*Whenever you have a number behind the decimal point*1585

*and at the end of a number, you can add as many 0s as you need.*1590

*We are going to add this.*1595

*4 plus 4 is 8; 8 plus 0 plus 0 is 8 point...*1597

*6 plus 3 is 9; plus 3 is 12; 7, 8, 9, 10.*1605

*1 there; 0 there; 3, 4, 5, 6.*1614

*My surface area when I add the lateral area... here is my base and my other base.*1619

*When you add lateral area plus the base plus the base,*1631

*we are going to get our surface area which is 602.88.*1635

*Our units is inches; area is always squared.*1642

*That is the surface area of this cylinder.*1649

*That is it for this lesson; thank you for watching Educator.com.*1655

1 answer

Last reply by: Maryanne Stevens

Fri Jan 29, 2016 6:38 PM

Post by Maryanne Stevens on January 29, 2016

Hello,

I think the last problem you forgot to multiply by 10 in regards to finding LA. You only did (2X3.14X6) which is 37.68, however you didn't multiply this by 10, which should be 3,768 inches square, then add the area of the two bases(circles in this case). I paused your video and completed the problem. ,y answer was 3,768 inches square + 2(113.04)inches = 3,994.08 inches square.

Am I right, or are you right?

Thank you

1 answer

Last reply by: Khanh Nguyen

Sat Jul 25, 2015 12:37 AM

Post by Milan Ray on April 15, 2014

What are the circles for!

0 answers

Post by Daniel G Swedlund on August 31, 2013

I think the latin number for 9 = IX. While XI = 11.

0 answers

Post by Arpana Duggal on July 2, 2012

this chapter isnt 11 its 9!