Geometry Volume of Pyramids and Cones

Welcome back to Educator.com.0000

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

Remember that volume is how much space the solid is taking up--how much space is inside the actual solid.0012

To find the volume of a cone, it is 1/3 times the area of the base, times the height.0018

Remember that, for the area of the base, we are going to use capital B; so whenever you see capital B, that is for the area of the base.0027

Lowercase b is just going to be the segment base; capital B is for the area of the actual base of the solid.0037

The volume of a cone is 1/3 the area of the base times the height.0046

Now, let's say that this cone has a radius of 3 centimeters, and has a height of 5 centimeters.0055

To find the area of the base for that, because the base of a cone is a circle, we know that the area of the base is πr².0076

This is 1/3 times πr² times the height: 1/3 times π times 3² times 5 for the height.0115

This is 9π times 5; I can go ahead and divide this number by 3, because this is over 1; so then, this simplifies to 3 times π times the 5.0139

So, this will be 15π; now, to simplify this out, you can just use your calculator (I have a calculator here on my screen):0141

it is going to be 15 times π is 47.12; the units will be in centimeters, and for volume, it is always cubed, remember.0152

So, this cone has a volume of 47.12 centimeters cubed; that is the volume of a cone, 1/3 times the area of the base times the height.0169

For a pyramid, it is actually the same exact formula: it is 1/3 times the area of the base times the height.0184

Now, if you were to base the formula on what kind of base of the solid you have, then it would have a different formula.0190

But because we can just say that it is every base, no matter what the base is,0201

no matter what type of polygon you have as the base, it will always be 1/3 area of the base, times the height.0207

Base for this, because we have a rectangle, is going to be the length times the width, and then times the height.0221

So, let's say I have...this is 6; this is 5; and my height is 10, and let's say this is all in meters.0233

Then, it is going to be the length, which is 6, times the width of 5, times the height of 10.0254

Now again, you can just go ahead and divide this and simplify this: 6 divided by 3 is 2, so it would be 2, times the 5, times 10.0265

And that is 10 times 10, is 100; my units are meters cubed.0278

And that is how you find the volume of the cone and the pyramid: 1/3 the area of the base, times the height.0288

For our first example, we have a pyramid, and the base of this pyramid0297

is a regular pentagon with the area of 20 kilometers squared; find the volume of the pyramid.0303

So, this base, the pentagon, is 20 kilometers squared, and we have to find the volume; so then, the volume is 1/3 area of the base times the height.0311

Now, we don't have to solve for the area of the base, because it is already given to us: so it is 1/3 times 20 times the height of 9.0327

Let's divide 9 and 3; that is 3, times 20, which is 60 kilometers cubed; that is the volume of this pyramid.0342

OK, for the next example, find the volume of the solid: here we have a cylinder, and we have a cone on top of the cylinder.0362

We are going to have to find the volume of each separately, and then we are going to add them together.0374

Now, if you notice, this solid right here, this cone, is not a right cone, because the vertex and the center of the base are not perpendicular.0379

This is called an oblique cone; to find the volume of this cone, it is actually the same exact formula,0393

but you just have to be careful with the height, because the height is no longer going to be from the vertex to the center of the base.0411

Just make sure that you remember that, for the height, it has to go from the vertex down to a point on the base, perpendicularly.0420

So then, this would be the height of this oblique cone.0431

Now, to find the volume of a cylinder, remember: it is going to be the area of the base, times the height.0435

For both solids, the base is going to be a circle: the radius is 5; 1/3πr² for the base...5 squared...the height is 7;0448

so, this is for the cone, and then this right here is for the cylinder: volume is, again, πr²,0470

so that is π(5)², times the height of 15.0490

When I add them together, I can just say that the volume of the solid is then going to be 1/3 times 25π0500

(I just squared this) times the 7, plus, again, 25π, times 15.0510

So here, this is the volume of the cone, and this is the volume of the cylinder.0524

I am going to use my calculator: I'll just punch it in...divide by 3...so the volume of the cone is 183 and 26 hundredths,0534

plus the next one: 1178.10; and then, when I add them together, cone plus the cylinder, I get 1361;0555

and the units are in centimeters, and volume...so it is cubed.0604

Make sure, when you have some kind of shape like this, that you recognize that this is a cone; this is a cylinder;0614

find the volume of each, and add them together, and that is going to be the volume of the whole solid.0620

A third example: Find the volume of the pyramid: here our base is a triangle; this is our height.0629

Now, what do we have here? To find the area of the base (because, again, my formula for volume0641

of this solid is going to be 1/3 the area of the base times the height), I am going to take this triangle;0652

I am going to re-draw it so that it is easier to see: this is 11 meters, this is 11, and this is 5.0668

Now, we don't have the height here, but I know that this is an isosceles triangle, so if I do that,0681

I can use half of this to make it a right triangle, and to use the Pythagorean theorem: so this is going to be 2.5; here is 11;0694

I am going to use those two to find this unknown.0704

So, if I call this a, it is going to be a² + 2.5² = 11².0709

And this is the Pythagorean theorem: a² + b² equals hypotenuse squared.0719

Just use your calculator, and get a²...you square this, and you square this;0731

you are going to subtract this from this, and I get this; and then, you are going to take the square root of that, and a is around 10.7.0743

So again, I had to find this because, to find the area of this base right here, I need the height.0773

I don't have the height, so to find it using the Pythagorean theorem, I am going to do a² + b² = c².0782

And then, that is going to be 10.7 for the height.0792

Then, the area of the base is going to be 1/2 base, which is 5, times height, which is 10.7 (what we just found).0799

And then, we get 26.something if we round that number; so then, here I am just going to plug this into the formula: 26.20.0817

And then, the height--do I have the height of the solid?0837

Let's see, this is my height of the solid; I don't have it, so I have to look for it.0842

Now, again, since this is 11, and this is 16, we can use this triangle (it is a right triangle), and using the Pythagorean theorem, I can find this height.0849

So, I am going to say h² + 11² = 16², the hypotenuse squared.0862

My height squared is 135; take the square root of that; h is 11.62.0884

I am going to include that into my formula: 11.62, and then just go ahead and use your calculator for the rest of that.0898

Multiply it by 6.78, and divide that number by 3; my volume is going to be 103.72; my units are meters cubed.0907

So again, this one took a little extra work, because we were missing some measures.0931

The height wasn't given to us; we had to find this height, so we used the Pythagorean theorem to find that height.0937

And then, to find the area of the base, this triangle, since the area of a triangle is 1/2 base times height, we had to find the height.0944

And we did that by the Pythagorean theorem: cutting this side in half, since it is an isosceles triangle, and then using this.0956

We found the area of the triangle; and once you find all of those unknown sides, just go ahead and plug them into the formula.0972

And use your calculator to solve it out.0979

For the fourth example, we are going to find the volume of the octahedron.0984

Now, we know that an octahedron is a regular polyhedron with 8 sides.0989

So, I have four sides up here, and I have four sides for this bottom part.1001

Let's break this up into two solids, two pyramids.1008

Now, because it is a regular solid, I know that all sides, all of the edges, have a measure of 10 centimeters.1011

This right here is the slant height; it is not the height of the actual solid.1021

So again, let's break this up into two pyramids, and then we are going to just add them together1027

(or multiply them: if we find the volume of one of the pyramids, let's say the top half--1035

the top half and the bottom half are exactly the same, so we could take this volume and then just multiply it by 2 to find the whole thing).1040

Again, I am just going to find the volume of the top half, the pyramid of the top part of it.1055

So then, here is the base; it is regular, so all of the sides are going to have a measure of 10 centimeters.1061

And then, something like that...I'll just do this one over...1071

Now, I need to find the actual height of the solid, from here all the way down to here, in the center.1091

I know that (and this is the slant height) the slant height has a measure of 8.7.1103

This base right here is 10; this is 8.7; let me just draw these sides blue, so that you know that it is this right here,1115

with this a side, the red side, the slant height; so this is 8.7; this is 10, because all of the sides are 10.1135

Then, half of this is going to be 5 centimeters long.1146

So then, using the Pythagorean theorem, this became 8.7; this is 5, because it is half of a side; so this is 5; this is 8.7.1152

Then, what is this right here--isn't it also 10?1169

So then, the height of that is going to be 10, because in a triangle, if we have 5 and then 8.7, then the hypotenuse is 10.1172

So, it is all going to be the same; the height is also 10.1184

The volume of this pyramid is 1/3 the area of the base, times the height.1191

So, it is 1/3...area of the base is 10 times 10, because it is a square...and then the height is 10.1201

Now, again, I am taking this whole thing and cutting it in half to find the volume of the upper part.1222

I am now going to take all of this, whatever that is, the volume, and then multiply it by 2 to find the area of the whole thing.1229

The volume of the solid (this is the volume formula) is going to be 1/3 times the area of the base, times the height,1240

and then times 2, because it is as if I am taking this whole volume, and then adding it to this volume;1259

and that is just times 2, because they are the same.1268

So, 10 times 10 is 100, times 10 is 1000; 1/3 times 1000 times 2 is 2000, divided by 3;1271

and then, using your calculator, the volume is 666.67 centimeters cubed.1295

And that is it for this lesson; thank you for watching Educator.com.1319