Geometry Surface Area and Volume of Spheres

Welcome back to Educator.com.0000

For the next lesson, we are going to go over surface area and volume of spheres.0002

First, let's go over some special segments within the sphere.0009

A radius, we know, is a segment whose endpoints are on the center and on the sphere--anywhere from center0014

(it has to be here), and then anywhere on the sphere--this is the radius.0027

A chord is a segment whose endpoints are on the sphere; it could be anywhere on the sphere.0032

So, it could be from here all the way to here; this is a chord, and this would be a chord; this would be a chord.0040

A diameter, we know, is a segment whose endpoints are on the sphere, but it has to pass through the center.0057

This is going to be a point on the diameter; it can go from here (that is the back side of the sphere),0065

and then passing through the center, and then to another point on the sphere; that would be the diameter.0074

And then, a tangent is a line that intersects the sphere at one point.0081

A chord, we know, is at two points; but this intersects only at one point.0088

So, it would be like this, where it is just intersecting at one point; that would be a tangent.0093

When it comes to intersecting a plane, a sphere can intersect at one point.0107

This is the plane, and here is a sphere; it is intersecting at one point.0114

Here, it is intersecting at a circle; so we can see the intersection where the plane and the sphere meet--it is this little circle right there.0119

That is where they are cutting.0131

And then, here, this is also a circle; but this is called a great circle, because it is the biggest circle0134

that can be formed from the intersection of a plane and a sphere.0146

It is basically passing through, intersecting, at the center of the sphere; and that is called the great circle.0150

This circle right here is called the great circle; this is just a circle, because the center is somewhere right here; it is not passing through the center.0157

But because this is passing through the center, it is the largest circle that can be formed with an intersection.0168

Now, with that intersection, this circle, the great circle, cuts the sphere in half; and that is called a hemisphere.0177

So, a half of a sphere, separated by a great circle, right at that great circle (half of the sphere)--this is a hemisphere.0188

To find the volume of this, you have to just divide it by 2.0200

OK, and then, the surface area of a sphere is 4πr².0206

This is the radius; it is just the area of the circle, times 4; it is 4πr², and that is surface area.0217

Remember: surface area is just the area of the whole outer part.0233

It is not the space inside (that would be volume); it is just the outside of it--that is the area, 4πr².0237

And then, to find the volume of the sphere, it will be 4/3πr³.0249

The surface area is 4πr², and the volume is 4/3πr³.0256

Let's go over our examples: Determine whether each statement is true or false.0266

Of a sphere, all of the chords are congruent.0271

We know that that is not true, because, within the sphere, I can have a chord passing just from here to here;0275

that is a chord (as long as the endpoints are on the sphere); I can have it passing through from here,0289

all the way to here; that would be a chord; so, chords are not all congruent--they could be, but they are not all congruent.0298

So, this would be a false statement.0308

In a sphere, all of the radii are congruent.0313

We know that this is true, because, whether the radius is going from here to here, or from here to here,0319

from here...because they are always from the center to a point on the sphere, these are all congruent; that is true.0328

The longest chord will always pass through the sphere's center.0342

Now, in order for it to be a chord, it just has to have the endpoints on the sphere.0347

If it passes through the center, we know that that is called the diameter; but that is also a chord, because the endpoints are on the sphere.0352

A diameter is a special type of chord; so this is true, because we can't draw a longer chord that doesn't pass through the center.0364

The next example: we are going to find the surface area of the sphere.0378

Here, again, for surface area, we are finding the area of the outside part--how much material is being used on the outside.0384

The surface area is 4πr², 4 times π...the radius is 5...squared; so this is going to be 4π times 25.0396

4 times 25 is 100, so it is 100π; or because we know that π is 3.14, 100 times that will be 314.0417

And the units are inches...this is surface area, so it is squared; the surface area of the sphere is 314 inches squared.0431

OK, the next example: Find the volume of a sphere with a diameter of 20 meters.0447

We know that a diameter is twice as long as the radius--from here, all the way going through, is the diameter;0456

and that is 20; so the radius is going to be half of that, which is 10 meters.0463

To find the volume, it is 4/3πr³, so the radius is 10, cubed...4/3π...10 cubed is not 30;0470

it is 10 times 10 times 10, which is 1000; so if I multiply 4 times 1000, is going to be 4000 times the π, over 3.0494

And you can just simplify that with your calculator: it is 4000 times π, divided by 3; and for the volume, I get 4188.79.0513

My units are meters...be careful here; it is not squared, because we are not dealing with area;0536

we are dealing with volume, how much space is inside this sphere; and that is cubed.0541

Now, the fourth example: we are going to find the surface area and the volume of the solid.0559

Here, we have a cylinder, and then let me just finish this off here like that, just so that you can see the cylinder.0565

And we have part of a sphere; now, let's see if this is a hemisphere (a hemisphere, remember, is half a sphere).0575

Here, the radius is 6, and this is also the radius of 6; we know that that is the radius, because that is the same measure.0589

So then, this has to be a hemisphere; now, be careful--just because you see part of a sphere doesn't automatically mean that it is a hemisphere.0597

It doesn't mean that it is half the sphere; here, I had to make sure, because,0605

from the center all the way from each side (to the point on the sphere), the radius has to be the same; then, that makes this a hemisphere.0613

So, to find the volume of this hemisphere (h for hemisphere), it is going to be the volume of a sphere, 4/3πr³;0625

and then, I am going to multiply this by 1/2, or divide the whole thing by 2, because it is half of a sphere.0645

That would be the volume of a hemisphere: just the volume of the whole thing, divided by 2.0656

And then, I have to find the volume of this cylinder, because it is the cylinder with the hemisphere.0664

So, when I find the volume of the cylinder (I'll write c for cylinder), that is the area of the base, πr², times the height, which is 8 there.0671

There is the formula for the hemisphere and the cylinder; and then I have to add them together.0689

Let's first find the volume of the hemisphere: 4/3π...the radius is 6, cubed; and then multiply that by 1/2.0698

OK, I am going to go ahead and multiply these fractions together: it is 4...and you can just go ahead0714

and start punching it into your calculator if you want to...here, this is 4, over 3 times 2 is 6, π, times 6 cubed.0720

Times the π...times the 4...and divide that by this...if it makes it easier for you, you can just find the volume0742

of this whole sphere first, and then just divide that by 2; I just went ahead and multiplied the fractions.0756

It doesn't really matter for now; just make sure that you cube this before multiplying; you can't multiply all of this with the 6,0764

and then cube it, because with Order of Operations, remember: you have to always do an exponent before you multiply.0775

I get 452.39; this is the volume of the hemisphere; units are cubed for volume.0783

And then, I have to find the volume of the cylinder, π...my radius is 6, squared, times the 8.0801

It is π, times 36, times 8; on my calculator, I do π times 36 times 8, and I get 904.78.0819

So then, I take this; I have to add it to my cylinder; add those two numbers together, and I get 1357.17; this is centimeters cubed.0848

The whole thing is centimeters cubed; there is my answer for the volume of this whole solid.0874

That is it for this lesson; thank you for watching Educator.com.0884