Basic Math Surface Area of a Prism

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

Surface area is when you are finding the area of all the sides of the solid.0008

For the prism, we are going to go over rectangular prism and triangular prism.0017

No matter what solid you are looking for, surface area always measures the area of all the sides.0022

Here the rectangular prism, imagine if you have a box and you are trying to wrap the box.0032

It is a gift for someone; you need to wrap it with wrapping paper.0039

All the sides of the box, all the sides that you are covering up, is going to be the surface area.0043

If we were to cut this up and unfold it, it is flat two-dimensional picture of what the box looks like.0055

That is called a net; this right here is a net.0067

These are all the sides of this rectangular prism.0072

If we unfold it up, it is going to look like this.0075

Here we have four sides right here.0080

We are going to call those lateral sides, lateral faces.0083

These two up here, we are going to call those the bases.0094

This top and the bottom are going to be our bases.0104

Surface area is going to find the area of each side.0115

You are going to find the area of this side, this side, this side, all six sides.0119

You are going to add them together; that is the surface area.0125

For lateral area, you are finding the area of all the lateral faces only.0130

Just these four and not the bases.0138

Lateral area is the area of all the faces except for the bases.0142

Just to demonstrate a little bit further on the net, if you have this, this is a rectangular prism.0151

If you were to cut it and unfold it, you are going to get a rectangle.0163

Here because it doesn't have the base, we are missing the base, this side and this side.0172

This would just be the measure of the lateral faces, the lateral area.0177

To find this lateral area, we know that this top part right here is the base.0185

If you find the measure of each of these sides of the base, that would be like this.0198

Again if we are going to cut it, let's say we are going to cut it right here.0211

We unfold it; it looks like this.0218

This right here would be the sides right here.0222

The cut is made right here; here is the cut; here is the cut.0234

To find the lateral area, we need to find the area of this rectangle.0244

We know that that is going to be the length times the width.0253

It is all of this times all of this.0257

The area here is length times the width.0262

How do we find the length and the width?0273

When you are given this, this side right here, the length, we are going to call this L for length.0276

It is actually going to be the perimeter of the base.0284

Because remember again it is all these four sides, all of those four sides.0287

Then the length is actually the perimeter of the base.0293

Here we know that that is just the width, the height.0305

Again to find the lateral area, you are going to find the area of this rectangle.0311

But this right here, to find the length of that, it is going to be the perimeter.0317

This one right here, this is side one; this is side two, side three, and side four.0325

This is going to be side one, side two, side three, and side four of the base.0332

It is the perimeter; that is how you would find the lateral area.0339

Lateral area is perimeter of the base times the height.0346

Or you can think of that as the width; the height which is this right here; the height of that solid.0358

Of course to find the surface area then, we are just going to add on the two bases.0367

The surface area is just going to be the whole lateral area, this whole thing.0373

Then we are going to just add on the area of the base.0379

Because we have two of them, we are going to just take that and multiply that by 2.0389

Lateral area, perimeter of the base times the height, is going to give you this, the area of this whole rectangle.0395

Surface area, because surface area is the sum of the areas of all the faces including the bases,0403

we are just going to take the lateral area and then just add on the area of this base right here0410

and then another area of the base which is down here; those two bases.0418

Let's just do a few examples; find the surface area of the rectangular prism.0429

Here again surface area, remember we are just finding the area of all the sides of this prism.0436

It is like... say this is face one; this is face two.0445

The front is face two; let's say this is face three.0460

The back is face four; five and six.0463

We have all these faces; there is five and six.0468

When it comes to surface area, regardless of the bases, we are just finding the area of all the sides.0477

We are just going to add them together.0484

Let's say face one first.0489

To find the area of face one, we are going to find the area of this top side right here.0492

That is this side times this side.0498

That is 10 times... what is that... 4 because that is the same as that; 10 times 4.0502

The length times the width of that face is 10 times 4 which is 40 centimeters squared.0511

Face two is this front... let's say that that is the front; face two.0523

That is 10 times this right here which is 6.0531

10 times 6 which is 60 centimeters squared for area.0538

Face three is the bottom side; the bottom side, length times width is 10.0546

Times the 4; that is 40 centimeters squared.0552

Face four, let's say that is the back.0562

That is this side times this; length times width.0566

This is 10 times 6; that is 60 centimeters squared.0573

If you say this is the side right here, length times width, 4 times 6, 24 centimeters squared.0585

The other side, we know that is the same as this side, 4 times 6.0599

We have the area of all six sides.0612

There is one, two, three, four, five, six; we can just add them all together.0616

40 plus 60 plus 40 plus 60 plus 24 plus 24; this is surface area.0628

This will be 100; this will be another 100.0656

It is 100 plus another 100 plus 48; the surface area is 248 centimeters squared.0661

All I did here was just find the area of each side, all six sides, and add them together.0689

That is surface area.0698

Of course we can find the measure of the lateral area and then find the base.0700

We can add those together; that would be a faster way to find the surface area.0709

But just so you understand, surface area, it is just finding the area of all the sides and then just add them all together.0714

The next one, here for this cube, we are going to actually find the lateral area and then the surface area.0725

Cube, we know all the sides are congruent.0732

Each side, each face is going to be a square.0737

If this is 5, then this side is going to be 5.0741

This side is going to be 5; all the sides are 5 inches.0744

For the lateral area, remember we have to first label the bases.0750

Let's say that the top and the bottom sides are going to be the bases.0755

Those are my bases.0771

That means to find the lateral area, I want to find the sum of0774

all the lateral faces, all the faces that are not red.0781

To do that, let me just demonstrate this cut again.0785

Let's say I am going to cut it right there; then I unfold it.0792

I am going to have one, two, three, four.0798

I am going to have this long rectangle.0811

This is where I cut; that is my cut.0815

To find the area of this, it is just one long rectangle.0821

It is just length times width.0828

This right here remember is this right here.0833

This is this; that is that; this is the fourth.0837

To find the length of this whole thing right here, this long rectangle here,0848

I just need to find the perimeter of this base.0853

If this is 5, then this is 5; this is also 5, 5, and 5.0859

The perimeter of this, 5, 10, 15, 20, the perimeter is 20 inches.0869

That means this side right here is 20 inches.0875

This height of this prism is we know 5.0885

This is also going to be 5.0889

To find the area of this whole thing, it is going to be 20 times 5, length times the width.0895

Lateral area is 20 inches times 5 inches which is 100.0904

For area, anytime you are finding area, whether it is lateral area or just area of a rectangle, inches squared.0917

To find the surface area, surface area is just all the sides.0933

Lateral area is just the four sides; that is not including the bases.0939

But we have two extra sides; those are the bases.0942

I just need to find the area of each of these bases.0947

Let's do that is in red; the base is going to be 5 times 5.0952

5 times 5; it is 25 inches squared.0963

I have two of them; base one; the bottom one, base two.0972

I just need to multiply this number by 2.0979

The area of my bases is 2 times 25 which is 50 inches squared.0982

All I have to do to find the surface area is...0994

since I have the area of all these sides right here, the four sides,1001

I just need to add on my two extra sides.1005

It is 100 inches squared plus 50 inches squared.1008

My surface area just becomes 150 inches squared.1019

For the third example, we are going to find the surface area of a triangular prism.1034

Again surface area, we are just finding the area of all the sides and adding them all together.1039

We know that this front side, the triangle, and the back side,1054

the one that is opposite, parallel, and congruent, are the bases.1061

The other three sides are my lateral faces.1068

This is a little bit different than the rectangular prism1073

because the rectangular prism, we are going to have six sides.1077

For this, we only have five because we have the three lateral faces which is this right here.1080

That is one face; then the other side, this face right here; and then the bottom face.1089

There is three; including my two bases, it will be five sides.1099

But still same thing, to find the surface area, we just have to find the area of all the sides.1106

The lateral area remember is still going to be the same.1114

This right here, this side, it is still a rectangle.1121

All of this is a rectangle.1130

If I am going to cut it right here, let's say I am going to cut it right here.1134

Then I am going to unfold it.1141

I am still going to get... here is a rectangle here.1145

There is this back side, rectangle; here is the bottom side.1152

This is where I cut it, right here.1163

Since this right here is 10, this is going to be 10.1172

We just have to find this area of the rectangle.1178

This is going to be my lateral area because this is my three sides1181

that are not including my bases; not including my triangles.1184

Here this side right here is this side right here; that is 5.1194

This next side right here is... it is this, then this.1204

That is also 5 because if you see here, they are congruent.1211

They are the same; that is also 5; it goes one, two, and three.1214

This right here is 6 meters.1222

To find the length of this right here, this side, it is still the perimeter of the base.1228

Remember the ones that we did, the rectangular prism.1233

It is always going to be the perimeter of the base.1236

Our base is a triangle; it is still the perimeter because we cut it here.1240

It went 5, 5, and 6.1244

We know this is 10; so we can find the lateral area.1252

This 5 plus 5 is 10 plus 6 is 16 meters.1259

To find the lateral area, I just multiply base times height or length times width.1267

10 times 16; my lateral area is 160 meters squared.1272

I want to find my triangle, my area of my base.1286

The area of a triangle is 1/2 base times the height.1295

1/2, my base is 6, my height is 4.1303

Remember the base and the height of the triangle have to be perpendicular to each other.1310

This is 6; this is 4.1313

Half of 6 is 3; times 4 is 12; 12 meters squared.1316

This is just the area for one of my bases.1327

I have two of them, the front and the back.1330

My bases, I have to multiply that by 2.1335

Or you can just do 12 plus 12; that is 24 meters squared.1342

To find my surface area, it is going to be the three sides here plus the two bases.1347

The three sides together is 160.1358

Plus my two bases which is this number here, 24.1361

It is going to give me 184; my units squared, meters squared.1370

Again surface area, you are just finding the area of all the sides.1384

Then you are going to add them together.1388

You can just find the area of all the lateral faces which is called the lateral area.1391

Then add on the areas of your bases.1399

That is going to give you surface area.1401

That is it for this lesson; thank you for watching Educator.com.1405