Chapter 14 Lesson 5
Solids with Circles
Surface Area of a Sphere
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The next solid is perhaps
the most important of all.
More important than cube, box,
prism, pyramid, can or cone?
Hard to imagine.
Sphere
You mean ball?
Why is it so important?
Think about it. From golf ball
to a planet, they are all related.
Sphere
One of my favorite spheres.
How many edges,
vertices, and faces in one?
0 Edges
0 Vertices
1 Face
Well, this is easy.
No vertices, no edges, 1 face.
0 Edges
0 Vertices
1 Face
Yes. Spheres have
1 face and it's not a circle.
1 oval
You're kidding me. An oval
is the face for a sphere?
1 oval
Yuppers. Lay it flat to see it.
In a way, the next
surface area is the easiest.
Just 1 eensy, tinsy formula.
I'll show it to you.
4 Pi x radius
2
That looks complicated.
4 Pi x radius
Not really. Radius
is squared, just like area.
2
4 Pi x radius
How do you say it?
2
4 Pi x radius
2
It's 4 pi, r squared.
I'll get my favorite radius.
4 Pi (1)
2
1 what? Foot, inch, mile?
4 Pi (1)
2
Mile would be really big.
We'll go with a ball that has a
radius of 1 ft. That's 2 feet across.
4 Pi (1)
2
So, your ball is 2 feet across.
How much material
would it take to make one?
4 Pi (1)
2
Not very big. I say 4 square feet.
4 (pi)(1) = 12.6
12.4 square feet?
Oooh, that's pretty big.
sq ft
4 (pi)(1) = 12.6
sq ft
That's with 3.14 as pi.
Do the same thing for 3 feet.
4 Pi (3)
2
So, your ball is 6 feet across.
How much material this time?
4 Pi (3)
2
Almost as tall as I am.
I'm guessing 60 sq ft.
4(3.14)(9) = 113.4
sq ft
Woe, twice as much as 60.
How do you
know these so fast?
4(3.14)(9) = 113.4
sq ft
Well, there is a little secret.
I have a number for 4 pi.
4 Pi (1)
2
4 pi is the same, no
matter what the radius is.
4 Pi (1)
2
So, what does that equal?
4 Pi R
2
12.6 R
2
4 x 3.14 is 12.6, so
you can use that instead.
4 Pi R
2
12.6 R
2
Do 1 more with that.
4 Pi R
2
12.6 (10)
2
Let's say the radius is 10 feet.
That's what it is for a tank.
4 Pi R
2
12.6 (10)
2
So, it's 12.6 x 100. I got that.
12.6 (100)
1,260 sq ft
1,260 square feet? That's huge.
If you build a round tank, you
know it takes alot of material.
4 Pi R
2
12.6 R
2
Which one should I remember?
4 Pi R
2
12.6 R
Either one, as long as you
remember what it stands for.
2
Sphere
So, was it really pretty easy?
Mostly, but I can't start shooting
a sphere in basketball. It's a ball.
Qs
What's the formula for
surface area of a sphere?
4 Pi x radius
2
That's how much material
covers the surface of a ball.
4 Pi R
2
What's the number formula
for surface area of a ball?
4 Pi R
2
12.6 R
2
4 x 3.1 is 12.6, so
you can use that instead.
Chapter 14 Lesson 5a
Solids with Circles
Volume of a sphere.
Sphere
Finally. The last solid.
What did you use to find
the volume of a pyramid?
6
6 feet
fe
et
............
height is 6 ft
A pyramid is 1 third of it's box.
6
6 feet
fe
et
............
height is 6 ft
What did you use to find
the volume of a cone?
Cone
A cone is 1 third of it's can.
Cone
I have some bad news for you.
Sphere
Sphere has nothing to do
with any other formula.
That is 1 generic looking ball.
Where did you get it? The trash?
Welcome to my art world.
It kind of looks like a ball.
Well, what's the new formula?
I'll tell you this. Sphere has
nothing to do the other formulas.
Not even Pi? It has to have Pi.
Well, you know it has Pi in it.
Ah HA! I was right. It has Pi.
Does it divide by 3 like cone?
Does it have 4
in it like surface area did?
Right again!!
It does have 1 third in it.
Darn! You are good.
It does have the number 4 in it.
Ok, here's the formula!!
3
4
Pi
x
R
3
Oh, that is different.
R to the 3rd power.
How do you say it?
3
4
Pi
x
R
3
4 thirds, Pi, R cubed
Time for a problem.
3
4
Pi
x
R
3
4 thirds, Pi, R cubed
Ok, what's the radius now?
Let me guess. I know.
3
4
Pi
x
R
3
3
4
3.1
x
1
3
I knew it was 1 foot.
Well, I got the 1st step.
3
4
3.1
x
1
3
4
3.1
3
I cubed it and multiplied.
You can finish it.
3
4
3.1
x
1
3
4
3.1
3
Thanks. You know 4 x 3.1.
Just divide that by 3.
4
3.1
=
12.4
3
Are you going to round
it or use the decimal?
3
4
3.1
=
12.4
3
3.133
3
cubic feet
3.1? I thought it would be more.
4
3.1
=
12.4
3
3.133
3
cubic feet
Looks like the number for Pi.
4
3.1
=
12.4
3
3.133
3
cubic feet
Hey, it really does.
4
3.1
=
12.4
3
3.133
3
cubic feet
One more reason Pi is amazing.
I'll make another problem up.
3
4
Pi
x
R
3
Before you go on, can you
use a number like 12.6 did?
3
4
Pi
x
R
3
You got me. I did the math.
Here's the new formula.
3
4
Pi
x
R
3
4.2 x R
3
I'll take 4.2 anyday. 100% easier.
3
4
Pi
x
R
3
4.2 x R
3
You know that's rounded.
Ok, time for a bigger ball.
3 ft
3 feet is pretty big.
You're going to make
me remember my cubes.
3 ft
Of course. What's the
number formula?
4.2 x R
3
4.2 x 3
3
That means it multiplies 3, 3 times
4.2 x R
3
4.2 x 3
3
Come on. It wasn't that long ago.
4.2 x 3
3
4.2 x 27 = 113.4
Multiply it is
113.4 cubic something.
4.2 x 3
3
4.2 x 27 = 113.4
Feet. It's cubic feet.
4.2 x 27 = 113.4
Cubic Feet
Just kidding. I was lazy.
4.2 x 27 = 113.4
Cubic Feet
Well, it was alot to think about.
Geometry
So, we're done with geometry.
More than that, here's what it is.
Measurement
Range, Mode, Median
Graphs Probability
Geometry
Yeah, that was alot to do.
All to get you ready for algebra.
Do I even want
to know what algebra is?
Right now? No. But it is
pretty amazing stuff.
Qs
What is the formula for
the volume of a sphere?
4 Pi x R 3
3
4 thirds Pi R cubed
What is the number formula
for the volume of a sphere?
4.1 x R
4.1 R cubed
3
Chapter 14 Lesson 5
Solids with Circles
Short Lesson
Surface Area and
volume of a Sphere
Sphere
First, how many
vertices and edges in a ball?
0 Edges
0 Vertices
No vertices and no edges.
How many faces are there?
1 oval
An oval is the face for a sphere.
In a way, the next
surface area is the easiest.
We'll show you the formula.
4 Pi x radius
2
It's 4 pi, r squared.
Radius is squared, just like area.
We'll get our favorite radius.
4 Pi (1)
2
A ball has a radius of 1 ft.
That's 2 feet across.
Find the surface area.
What will multiply here?
4 (3.1)(1) = 12.4 sq ft
Multiply is 12.4 square feet.
Bigger than you thought.
Do the same thing for 2 feet.
4 Pi (2)
2
The ball is 4 feet across.
How much material this time?
Do the exponent and multiply.
4(3.1)(4)
(3.1)(16)
It ends up 3.1 times 16.
What will that equal?
4(3.1)(4) = 49.6
There is a little secret
to multiplying these.
sq ft
4 Pi (1)
2
4 pi is the same, no
matter what the radius is.
What does that equal?
4 Pi R
2
12.4 R
2
4 x 3.1 is 12.4, so
you can use that instead.
Here's 1 more problem.
4 Pi R
2
12.4 (10)
2
Let's say the radius is 10 feet.
That's surface area of a tank.
You know it's 12.4 x 100.
12.4 (100)
1,240 sq ft
1,240 square feet. That's huge.
If you build a round tank, you
know it takes alot of material.
4 Pi R
2
12.4 R
2
Which one should you remember?
Either one, as long as you
remember what it stands for.
Sphere
What did you use to find
the volume of a pyramid?
6
6 feet
fe
et
............
height is 6 ft
A pyramid is 1 third of it's box.
What did you use to find
the volume of a cone?
Cone
A cone is 1 third of it's can.
Sphere works the same way.
Sphere
You could try guessing this,
but, there is 1 key difference.
3
4
Pi
x
R
3
4 thirds, Pi, R cubed
It's R to the 3rd power.
We'll use 1 foot for the radius.
3
4
Pi
x
R
3
3
4
3.1
x
1
3
There's all the numbers.
Multiply it all out.
3
4
3.1
x
1
3
4
3.1
3
You know 4 x 3.1.
Just divide that by 3.
What's the answer?
4
3.1
=
12.4
3
4.133
3
cubic feet
There is a shortcut for this.
3
4
Pi
x
R
3
4.1 x R
3
You can use 4.1
instead of 4 thirds pi.
Round to 10ths if you need to.
3 ft
It helps to know your cubes.
What's the number formula?
4.1 x R
3
4.1 x 3
3
4.1 x R cubed.
What is 3 cubed?
4.1 x 3
3
4.1 x 27
R cubed makes 27.
What's 4 x 27?
Then add 1 10th of 27.
4.1 x 3
3
4 x 27 = 108
108 is alittle bigger than 100.
Ok, what's 0.1 x 27?
Add that in there.
4.1 x 3
3
4 x 27.1 = 110.7
Add another 2.7 is 110.7.
110.7 what?
What's the label for it?
4 x 27.1 = 110.7
Cubic Feet
110.7 cubic feet.
Volume is always cubic units.
Questions
What's the formula for
surface area of a sphere?
4 Pi x radius
2
4 pi, r squared
What's the number for 4 pi?
4 Pi R
2
12.4 R
2
Use 3.1 for pi makes 12.4.
Questions
4 Pi x radius
Name 2 ways it changes
for volume of a sphere.
2
4 Pi x R 3
3
Divide by 3 and radius is cubed.
What is the number formula?
4 Pi x R 3
3
3
4.1 x R
4 thirds pi is about 4.1 (rounded).
Chapter 14 Lesson 5
Solids with Circles
Practice #1
Surface Area and
volume of a Sphere
What's the formula for
surface area of a sphere?
4 Pi x radius
2
4 pi, r squared
What's the number for 4 pi?
4 Pi R
2
12.4 R
2
Use 3.1 for pi makes 12.4.
4 Pi x radius
Name 2 ways it changes
for volume of a sphere.
2
4 Pi x R 3
3
Divide by 3 and radius is cubed.
What is the number formula?
4 Pi x R 3
3
3
4.1 x R
4 thirds pi is about 4.1 (rounded).
Problems
5 in
What's the the surface area
of a sphere that's radius is 5 in?
12.4 R
2
12.4 (5)
2
12.4 (25)
What's 12.4 x 25?
(Remember, 4 25s are 100.)
12.4 R
2
12.4 (5)
2
12.4 (25) = 310
The ball has a surface area
of 310 square inches.
sq in
6 in
What's the the surface area
of a sphere that's radius is 6 ft?
12.4 R
2
12.4 (6)
2
12.4 (36)
Estimate it. What's 12.4 x 36?
12.4 R
2
12.4 (6)
2
12.4(36) = 446
The ball has a surface area
of 446 square inches.
sq in
Chapter 14 Lesson 5
Solids with Circles
Practice #2
Surface Area and
volume of a Sphere
What's the formula for
surface area of a sphere?
4 Pi x radius
2
4 pi, r squared
What's the number for 4 pi?
4 Pi R
2
12.4 R
2
Use 3.1 for pi makes 12.4.
4 Pi x radius
2
Name 2 ways surface area
changes for volume of a sphere.
4 Pi x R 3
3
Divide by 3 and radius is cubed.
What is the number formula?
4 Pi x R 3
3
3
4.1 x R
4 thirds pi is about 4.1 (rounded).
Problems
3 in
What's the equation for the
volume of a sphere with 3 in radius?
4.1 x 3
3
4.1 x 27
4.0 x 27
Round to 4. What's the volume?
4x3
3
108 cu ft
27
x 4
108
Rounded answer is 108 cu ft.
What if it multiplied 4.1?
4.1 x 3
3
108 cu ft
27
x 4.1
27
108
110.7
108 cu ft is 2.7 off.
10 in
What's the equation for the
volume of a sphere with 10 in radius?
4.1 x 10
3
4.1 x 1000
10 cubed is 1000.
What's the answer?
4.1 x 10
3
4.1 x 1000
4100
cubic inches
Volume is 4100 cubic inches.
Chapter 14 Lesson 5
Solids with Circles
Performance #1
Surface Area and
volume of a Sphere
2 ft
What's the formula for the surface
area of a sphere with radius of 2 ft?
(We'll use the number one.)
4 Pi R
2
12.4 R
2
Use 3.1 for pi makes 12.4.
Solve for R is 2.
4 Pi R
2
12.4 2
2
12.4 (4)
What's 12.4 x 4?
12.4 R
2
12.4 (2)
2
12.4 (4) = 49.6
The ball has a surface area
of 49.6 square feet.
sq ft
3 ft
Add a foot.
What's the surface area of
a sphere with radius of 3 ft?
4 Pi R
2
12.4 3
2
12.4 (9)
What's 12.4 x 9?
12.4 R
2
12.4 (3)
2
12.4 (9) = 111.6
The ball has a surface area
of 111.6 square feet.
sq ft
2 ft
What's the formula for the volume
of a sphere that's radius is 2 ft?
(Again, the number one.)
4.1 x R
3
4.1 x 2
3
4.1 times radius cubed.
What does it multiply?
4.1 x 2
3
4.1 x 8
2 cubed is 8.
What's the volume?
4.1 x 2
3
4.1 x 8
32.8
cubic inches
It has a volume of 32.8 cubic inches.
3 ft
Add a foot.
What's the volume of a
sphere that's radius is 3 ft?
4.1 x 3
3
4.1 x 27
3 cubed is 27.
What's the volume?
4.1 x 3
3
4.1 x 27
110.7
cubic inches
It has a volume of 110.7 cubic inches.
Chapter 14 Lesson 5
Solids with Circles
Performance #1
Surface Area and
volume of a Sphere
4 in
What's the formula for the surface
area of a sphere with radius of 4 in?
(We'll use the number one.)
4 Pi R
2
12.4 R
2
Use 3.1 for pi makes 12.4.
Solve for R is 4.
4 Pi R
2
12.4 4
2
12.4 (16)
What's 12.4 x 16?
12.4 R
2
12.4 (4)
2
12.4(16) = 198.4
The ball has a surface area
of 198.4 square feet.
sq ft
10 in
Add 6 inches.
What's the surface area of
a sphere with radius of 10 in?
4 Pi R
2
12.4 10
2
12.4 (100)
What's 12.4 x 100?
12.4 R
2
12.4 10
2
12.4(100) = 1240
The ball has a surface area
of 1240 square inches.
sq ft
4 in
What's the formula for the volume
of a sphere that's radius is 4 in?
(Again, the number one.)
4.1 x R
3
4.1 x 4
3
4.1 times radius cubed.
What does it multiply?
4.1 x 4
3
4.1 x 64
4 cubed is 64.
What's the volume?
4.1 x 4
3
4.1 x 64
262.4
cubic inches
It has a volume of 262.4 cubic inches.
10 in
Add 6 inches.
What's the volume of a
sphere that's radius is 10 in?
4.1 x 10
3
4.1 x 1000
10 cubed is 1000.
What's the volume?
4.1 x 10
3
4.1 x 1000
4100
cubic inches
It has a volume of 4100 cubic inches.