Basic Math 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 width0121

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 πr²; 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 πr².0291

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 πr² 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 time0371

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 label0388

plus the area of the base which is a circle, πr² 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, πr², all of this right here.0589

πr²; the area of the base is πr².0603

That is 3.14 times the radius which is 4².0613

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 πr² 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 πr²; π is 3.14; radius is 6².1065

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 point1163

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 be1319

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 is1418

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 πr².1448

π is 3.14; the radius is 6; 3.14 times 6².1453

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 point1585

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