Geometry Volume of Prisms and Cylinders

Welcome back to Educator.com.0000

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

Volume is the measure of all of the space inside the solid.0011

We went over lateral area and surface area: remember, lateral area measures the area of all of the outside sides, except for the bases;0020

and then, surface area would be the area of all of the sides together, including the bases.0035

If you were to wrap a box, let's say, with wrapping paper, that would be surface area.0044

If you were to fill the box with something (let's say water or sand or anything--just filling it up), that would be volume.0051

That is going to be the volume of that box.0059

Now, the box itself is a prism; prisms, remember, are solids with two congruent and parallel bases.0065

They have to be opposite sides; they are two congruent and opposite bases.0078

It can be any type of shape; it can be a triangle; it can be a rectangle; it can be a hexagon, pentagon...whatever it is.0082

And the rest of the sides, the lateral sides (meaning the sides that are not bases) have to be rectangles; that is a prism.0091

This right here is a rectangular prism, because the bases are rectangles.0102

Let's say that we are going to name that the base; then, that means that the bottom is also the base.0111

The volume (meaning everything inside, the space inside) is going to be the area of the base, times the height--0125

the area of this right here, times the height of the prism.0140

When it comes to a rectangular prism, we know that the area of the base is the length times the width.0147

And then, the volume will be times the height; so length times width times height is for a rectangular prism.0157

Length times width measures the area of the base; any time you have a lowercase b, that just means the segment base--0168

the line segment, like maybe the measure of the bottom side or something; that is the base, lowercase b.0178

When you have a capital B, that is talking about the area of the actual base, the side base.0186

A capital B is the area of the base, times the height; and this is the volume.0194

For rectangular prisms, it would just be the length times the width; but for any type of prism, it is going to be capital B,0201

for base, the area of the base, times height; so this is the formula for the volume of a prism.0210

Now, the volume of a cylinder: remember: a cylinder is like a prism in that there are two bases, opposite and congruent and parallel.0220

But for a cylinder, the bases are circles: circle and circle.0232

Now, to find the volume, again, we are measuring the space inside.0241

So, if you were to take a can, and you were to fill it up with water or something, that would be volume.0247

How much water can that can take?--that would be volume.0252

The formula for this is actually the same as the formula for the prism: πr² is the area of the base;0263

that is the formula for the area of a circle (which is the base), times the height.0277

So, for a cylinder, volume is also capital B, for the area of the base, times the height.0285

So again, prism and cylinder have the same formula for volume.0295

Just think of capital B as the area of the base; whatever the base is, it is the area of that side, times the height.0300

The first example: we are going to find the volume of this prism.0313

Now, remember: when you have a prism, all of the lateral sides--meaning all sides that are not bases--have to be rectangular.0317

If you have a prism, and you know that it is a prism for sure, then the sides that are not rectangular automatically become the bases.0326

So here, I see a triangle; that would have to be the base, the side that is the base,0336

which means that the side opposite has to also be the same; it has to also be congruent; if it is not, then it is not a prism.0346

There is our base; to find the volume, remember, you are going to find the area of that base, times the height.0358

The area of this triangle, we know, is 1/2 base times height (and this is lowercase b, times the height)--is all volume.0370

This base and the height are 6 and 10; we know that those two are our measures, because they are perpendicular.0388

The base and the height for the triangle have to be perpendicular, which they are; that is what this means, right here.0395

So, it is 6 and 10; so 1/2 times 6 times 10, times the height of the prism (that is 4)...we just multiply all of those numbers together, and that is our volume.0400

Here, 1/2 times 6 is 3, so I just cross-cancel this number out: 3 times 10 is...so all of this together is going to be...30,0417

times the 4, is 120; our units are inches...for volume, it is going to be units cubed.0433

For area, we know that it is squared; volume is always going to be cubed.0447

Here, this is the volume of this prism; again, it is the area of the base, 1/2 base times height, times the height of the prism.0454

Be careful that you don't confuse this h with this h; this h is just the height for the base, for the triangle;0466

and this height is the height of the actual prism.0477

The next example: Find the volume of the cylinder.0485

Again, the formula is the same: capital B (for area of the base), times the height.0490

In this case, the base is a circle; the area of a circle is π (oh, that is not π) r², times the height.0500

πr...the radius is 4, squared, times 10 for my height...just to solve this out, to simplify this,0519

it is going to be 4², which is 16, times 10; that is 160π.0533

To turn that into a decimal, to actually multiply this out, 160 times π, you can go ahead and use your calculator.0541

I have mine on my screen; so it is 160 times π; so my volume becomes 502.65; my units are centimeters, and then volume is cubed.0548

So, there is the volume of my cylinder.0570

The next example: A regular hexagonal prism has a length of 20 feet, and a base with sides 4 feet long; find the volume of the prism.0577

Let's see, a regular hexagonal prism: "hexagon" means 6 sides, so that means the base of my prism is going to be a hexagon.0591

Let me try drawing this as best I can--something like that.0604

It has a length of 20 feet and a base (that is this hexagon right here) with sides four feet long; so each of these is four feet.0623

Make sure that you remember that it is regular, so it has to be equilateral and equiangular.0635

That means that each of these sides is going to be 4 feet long; find the volume of this prism.0640

Back to the formula: volume equals the area of the base, times height.0648

Now, to find the area of this base, it is a hexagon; so remember: to find the area of a regular polygon...let me just review over that quickly.0658

If I have a regular hexagon (that doesn't really look regular, but let's just say it is), it is as if I take this hexagon,0673

and I break it up into congruent triangles: I have 1, 2, 3, 4, 5, 6 triangles.0682

The area of each of these triangles is 1/2 base times height; so if this is the base, and this is the height (that is for this triangle),0690

and then I have 6 of them--so then, here is 1, 2, 3, 4, 5, 6--it is 1/2 base times height; multiply that by 6.0705

Well, let me just use a different color, just to show you that this is on-the-side review.0719

If I take the base (this is the base), and I multiply it by the 6, well, if this is the base, there are 6 sides.0725

Then, my base and my 6 become the perimeter; and then, my height of the triangle, this right here, is what is called the apothem.0739

And then, my 1/2...so then, the area of this whole thing, which is 1/2 base times height, times 6,0758

because there are 6 triangles, becomes 1/2 times the perimeter times the apothem.0766

So, this base is 1/2 times the perimeter times the apothem, and then times this height right here, the height of the prism.0774

Let's see, my perimeter is going to be (since it is a regular hexagon) four times...I have 6 sides, so...4 times 6 is 24;0786

and then, my apothem, remember, is from the center to the midpoint of one of the sides; so that is perpendicular.0809

That is my apothem; now, I don't know what my apothem is, so I have to solve for it.0825

So, I am going to make a right triangle; a right triangle is always the best way to find unknown measures.0831

I am just going to draw that triangle a little bigger, just so that you can see.0845

My apothem (that is this right here): if this whole side is 4, then I know that this right here has to be 2; it is half.0854

And then, since that is all I have to work with, let's see: if I had this measure,0866

I could use the Pythagorean theorem, a² + b² = c²; but I don't.0872

If you did have that, then you could use the Pythagorean theorem.0878

What I can do: I know that from here all the way around, passing through all 6 triangles, it is going to have a measure of 360 degrees.0882

I just want to find the measure of this angle right here: 360 divided by each of the 6 triangles is going to be 60 degrees.0901

That means that this right here, just one triangle, is going to have a measure of 60;0914

it is going to have a measure of 60; this is 60; 60; and 60, and so on.0923

If from here, all of this is 60, then I know that half of this right here has to be 30 degrees.0930

If this is 30, and this is 90, then this has to be 60; and that is 0...60 degrees.0941

So then, this becomes a 30-60-90 degree triangle; and again, I found out that this is 300950

by taking my 360, which is the whole thing, and dividing it by 6 (because there are 6 triangles).0958

So then, this triangle and this triangle, all 6 of them--each of them is going to have an angle measure of 60 degrees.0968

So, this is 60; but then I have to cut it in half again, so then, that is 30.0975

This becomes a special right triangle--remember special right triangles?--it is a 30-60-90 degree triangle.0981

The side opposite the 30 degrees is going to be n; the side opposite the 60 is n√3; the side opposite the 90 is 2n.0992

n is just the variable for the special right triangles; that is what I am going to use.1004

Now, what side is given to me? I have a 2 here--that is the side opposite the 30.1011

That is this right here; so I am going to make those two equal to each other: n = 2.1016

Well, if n equals 2, then what is the side opposite the 60? That is going to be 2√3.1024

Then, the side opposite the 90 is going to be 2 times n, which is 4.1029

OK, that means that the side opposite the 60 is 2√3; the side opposite the 90 is 4.1035

My apothem, which is this side right here, is going to be 2√3; my height of the prism (because we are back to this volume formula--1044

not the height of the triangle, but the height of the prism) is 20 feet.1063

So again, it is 1/2 times the perimeter, times the apothem, times the height.1071

To find the perimeter, remember (and all of this, 1/2Pa--that is for the area of the base): you are going to take the 4,1078

and multiply it by all of the sides; that is 24; to find the apothem, you just use special right triangles.1087

This is 2, so then this is 2√3, and then multiply that by 20; so I am going to have...1097

and then, you can just use your calculator for that...times that number, times...and I get 831.38 feet cubed for volume.1109

So, the volume of this prism is that right there.1139

And then, the fourth example: Find the volume of the solid.1147

For this, we have two prisms stacked like that; so to find the volume of this whole thing, I need to find the volume of this prism,1152

the one on the bottom; find the volume of this top prism; and then add them together.1165

I am going to say that this is prism #1 here; this is #1, and this is #2.1175

#1 is going to be...volume equals capital B (for the area of the base), times the height.1185

The base...this is a rectangular prism, so it is 10 times 10; the length times the width is 10 times 101196

(because it is the same; they are congruent) times the height, which is 4.1214

So, the volume of this first prism, prism #1, is 100 times 4, which is 400 meters cubed.1219

Then, I have to find the volume of prism #2; it is also a rectangular prism, so length times width times the height.1231

The length and the width are congruent, so this is going to be 6 times 6; and the height is 2.1248

I know that 6 times 6 is 36, times 2 is 72; that is meters cubed.1261

The volume of this solid is going to be prism 1, plus prism 2.1273

Prism 1 is 400 meters cubed, plus the 72 meters cubed, which is going to be 472 meters cubed.1280

Make sure that you add the two volumes together; you are not multiplying them,1296

because it is the volume of that whole thing, plus the volume of this whole thing.1303

You are adding them together--the total is the sum of them.1309

That is it for this lesson; thank you for watching Educator.com.1316