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### Surface Area and Volume of Spheres

- Special segments of a sphere
- Radius: endpoints are the center and on the sphere
- Chord: endpoints are points on the sphere
- Diameter: endpoints on the sphere and passes through center
- Tangent: a line that intersects the sphere at one point
- A plane can intersect a sphere in a point, a circle, or a great circle
- Hemisphere: half of a sphere separated by a great circle
- Surface area of a sphere = 4πr
^{2} - Volume of a sphere = 4/3 (πr
^{3})

### Surface Area and Volume of Spheres

The two endpoints of a chord are both on the sphere.

A tangent of a sphere can intersect the sphere at one or two points

The radius of the great circle of a sphere is ____ the same as the radius of this sphere.

- SA = 4πr
^{2} - SA = 4*3.14*4
^{2}

^{2}

- V = [4/3]πr
^{3} - V = [4/3]*3.14*5
^{3}

^{3}.

- SA = 4πr
^{2} - r = √{[SA/(4π)]}
- r = √{[188/4*3.14]}

In a sphere, a radius is always longer than a chord.

In a sphere, a chord is ________ equal or smaller than the diameter.

- V = [4/3]πr
^{3} - r =
^{3}√{[3V/(4π)]} - r =
^{3}√{[3*285/4*3.14]}

The radii of a hemisphere and a sphere are the same, then the surface area of the hemisphere is a half of the surface area of the sphere.

*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

### Surface Area and Volume of Spheres

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
- Special Segments
- Sphere
- Surface Area of a Sphere
- Volume of a Sphere
- Extra Example 1: Determine Whether Each Statement is True or False
- Extra Example 2: Find the Surface Area of the Sphere
- Extra Example 3: Find the Volume of the Sphere with a Diameter of 20 Meters
- Extra Example 4: Find the Surface Area and Volume of the Solid

- Intro 0:00
- Special Segments 0:06
- Radius
- Chord
- Diameter
- Tangent
- Sphere 1:43
- Plane & Sphere
- Hemisphere
- Surface Area of a Sphere 3:25
- Surface Area of a Sphere
- Volume of a Sphere 4:08
- Volume of a Sphere
- Extra Example 1: Determine Whether Each Statement is True or False 4:24
- Extra Example 2: Find the Surface Area of the Sphere 6:17
- Extra Example 3: Find the Volume of the Sphere with a Diameter of 20 Meters 7:25
- Extra Example 4: Find the Surface Area and Volume of the Solid 9:17

### Geometry Online Course

### Transcription: 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 center*0014

*(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 circle*0134

*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 4πr ^{2}.*0206

*This is the radius; it is just the area of the circle, times 4; it is 4πr ^{2}, 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, 4πr ^{2}.*0237

*And then, to find the volume of the sphere, it will be 4/3πr ^{3}.*0249

*The surface area is 4πr ^{2}, and the volume is 4/3πr^{3}.*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 4πr ^{2}, 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 4/3πr ^{3}, 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, 4/3πr ^{3};*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, πr ^{2}, 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 ahead*0714

*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 volume*0742

*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

0 answers

Post by Hakan Sisek on June 24, 2016

is their a latreal area foir shpere why or why not

0 answers

Post by Jonathan Aguero on March 22, 2013

how do I find the radius and height of cylinder knowing that the height is 4 times the radius being in which 301.593 is the surface area

0 answers

Post by Shoshana Coleman on February 19, 2013

What about the surface area?(in example 4)

0 answers

Post by JOAN CHEN on September 1, 2011

You forgot to find the surface area in Extra Example 4.