### Volume of Prisms and Cylinders

- Volume of a prism = area of the base × height
- Volume of a cylinder = πr
^{2}h

### Volume of Prisms and Cylinders

The volume of a prism = [1/2] * (area of the base) * height

Find the volume of the cube, each side of the cube is 5m.

- V = Bh
- V = 5*5*5

^{3}

The volume of a cylinder = (area of the base) * height

- radius = [1/2]diameter = 4in
- V = πr
^{2}*h - V = 3.14*4
^{2}*10

^{3}

If the a cylinder and a prism have the same area of the base, and the same height, then their volume are ____ the same.

Find the volume of this prism, the bases are trapezoids, EH = 4, FG = 8, EM = 7, BF = 12.

- Area of base = [1/2](EH + FG)*EM
- Area of base = [1/2]*(4 + 8)*7 = 42
- V = Area of base*height
- V = 42*12 = 504

Find the volume of the solid, the base of the prism is a rectangle.

- V of the prism = area of base*height
- V of the prism = 30*25*5 = 3750
- V of the cylinder = πr
^{2}*height - V of the cylinder = 3.14*6*6*20 = 2260.8
- V of the solid = V of the prism + V of the cylinder
- V = 3750 + 2260.8 = 6010.8

The base of a cylinder is always a circle.

All the lateral faces of a prism are rectangles.

The perimeter of the base circle of is 18, the height of the cylinder is 15, find the volume of this cylinder.

- P = 2πr
- r = [P/(2π)] = [18/2*3.14] = 2.87
- V = πr
^{2}*H - V = 3.14 * 2.87
^{2}* 15

*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

### Volume 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 0:00
- Volume of Prism 0:08
- Volume of Prism
- Volume of Cylinder 3:38
- Volume of Cylinder
- Extra Example 1: Find the Volume of the Prism 5:10
- Extra Example 2: Find the Volume of the Cylinder 8:03
- Extra Example 3: Find the Volume of the Prism 9:35
- Extra Example 4: Find the Volume of the Solid 19:06

### Geometry Online Course

### Transcription: 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: πr ^{2} 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) r ^{2}, 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 4 ^{2}, 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, a ^{2} + b^{2} = c^{2}; 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 30*0950

*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 10*1196

*(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

0 answers

Post by Jonathan Aguero on March 1, 2013

how is it possible to find the volume of a miss figured shape that is oblique