Transcript 12 6 12 7 Volume SA Spheres Similar

Assignment
• P. 842-5: 2, 3-11
odd, 12-20 even,
21-23, 28, 33-38
• P. 850-3: 1, 2, 3-21
odd, 24, 30
• Solids of
Revolution
Worksheet
Warm-Up
Draw and name the 3-D solid of revolution
formed by revolving the given 2-D shape
around the x-axis.
Warm-Up
Draw and name the 3-D solid of revolution
formed by revolving the given 2-D shape
around the x-axis.
Sphere
Hemisphere
Torus
12.6-12.7: Volume and Surface Area of
Spheres and Similar Solids
Objectives:
1. To derive and use the formulas for the
volume and surface area of a sphere
2. To find the surface area and volume of
similar solids
Sphere
A sphere is the set
of all points in
space at a fixed
distance from a
given point.
• Radius = fixed
distance
• Center = given
point
Exercise 1
What is the result of
cutting a sphere
with a plane that
intersects the center
of the sphere?
What 2-D shape is
projected onto the
plane?
Hemisphere
A hemisphere is
half a sphere.
The circle on the
base of a
hemisphere is a
great circle.
Investigation 1
In this Investigation, we
will discover the formula
for the volume of a
sphere. To do this we
need to relate the
sphere to a very
particular cylinder.
Sphere
Cylinder
Radius = r
Radius = r
Height = 2r
Investigation 1
In this Investigation, we
will discover the formula
for the volume of a
sphere. To do this we
need to relate the
sphere to a very
particular cylinder.
Notice that this is the largest possible sphere that could fill
the cylinder. This sphere is inscribed within the cylinder.
Investigation 1
Step 1: Rather than use the
sphere, we’ll use the
hemisphere with the
same radius, since it will
be easier to fill. So…fill
the hemisphere.
Step 2: Pour the contents
of the hemisphere into the
cylinder. How full is it?
Investigation 1
Step 3: Repeat steps 1
and 2. How full is the
cylinder?
Step 4: Repeat step 3.
How full is the cylinder?
What does this tell you
about the volume of the
sphere?
Archimedes Tomb
Archimedes was the first
to discover that the
volume of a sphere is
2/3 the volume of the
cylinder that
circumscribes it. He
considered this to be his
greatest mathematical
achievement.
Exercise 2
Derive a formula for the
volume of a sphere.
VSphere
2
  VCylinder
3
2
   r 2 h 
3
2
   r 2  2r 
3
4
  r3
3
h = 2r
Exercise 3
Derive a formula
for the volume of
a hemisphere.
Exercise 4
What is the extended ratio of the volume of
the cone to the sphere to the cylinder?
Volume of Spheres and Hemispheres
Volume of a
Sphere
Volume of a
Hemisphere
V  43  r 3
•
r = radius of the sphere
V  23  r 3
•
r = radius of the hemisphere
Exercise 5
Find the volume of each solid using the given measure.
1. d = 18.5 inches
2. C = 24,900 miles
Exercise 6
Find the volume of each solid using the given measures.
1. V =
2. V =
Investigation 2
Now we’ll find a
formula for the
surface area of a
sphere. To do this,
perhaps we should
use a net…
Or perhaps we’ll look
at it another way.
Investigation 2
Think of a sphere as
being constructed by a
whole bunch of
pyramids—I mean
bunch of them. The
height of each pyramid
would be the radius of
the sphere.
n = a whole bunch
h = radius of the sphere
B
Investigation 2
Let’s also say that each
of these pyramids is
congruent and has a
base area of B.
Thus, the surface area of
the sphere is:
S  B1  B2  B3 
 Bn
(Not a very useful formula)
B
Investigation 2
Furthermore, the volume
of the sphere should
be the sum of the
volumes of the
pyramids.
V  13 B1h  13 B2 h  13 B3h 
 13 Bn h
V  13 h  B1  B2  B3 
 Bn 
V  13 r  B1  B2  B3 
 Bn 
V  13 r  S
B
Exercise 7
Use the two formulas
below to derive a
formula for the surface
area of a sphere.
V  13 r  S
V  43  r 3
B
Exercise 8
Explain how the
unwrapped
baseball illustrates
the formula for the
surface area of a
sphere.
Exercise 9
Derive a formula for the total surface area of
a hemisphere.
SA of Spheres and Hemispheres
Surface Area of a
Sphere
S  4 r
•
Surface Area of a
Hemisphere
S  3 r
2
r = radius of the sphere
•
2
r = radius of the hemisphere
Exercise 10
Find the surface area of each solid using the given measure.
1. d = 18.5 inches
2. C = 24,900 miles
Similar Solids
Any two solids are similar solids if they are of the
same type such that any corresponding linear
measures (height, radius, etc.) have equal ratios.
– Ratio = scale factor
Exercise 11
Explain why any two
cubes are similar.
2"
4"
Exercise 12
Find the volume of a
cube with a side
length of 2 inches.
Now find the volume
of a cube with a
side length of 4
inches.
How do the volumes
compare?
2"
4"
Exercise 12
Find the volume of a
cube with a side
length of 2 inches.
Now find the volume
of a cube with a
side length of 4
inches.
How do the volumes
compare?
2"
2"
2"
2"
2"
2"
2"
4"
2"
2"
Volumes of Similar Figures
If two solids have a
scale factor of a:b,
then the
corresponding
volumes have a
ratio of a3:b3.
Similarity Relationships
For two shapes with a scale factor of a:b, each of the
following relationships will be true.
Exercise 13
A breakfast-cereal manufacturer
is using a scale factor of 5/2
to increase the size of one of
its cereal boxes. If the volume
of the original cereal box was
240 in.3, what is the volume of
the enlarged box?
Exercise 14
Pyramids P and Q are similar. Find the scale
factor of pyramid P to pyramid Q.
V = 1000 in3
V = 216 in3
Assignment
• P. 842-5: 2, 3-11
odd, 12-20 even,
21-23, 28, 33-38
• P. 850-3: 1, 2, 3-21
odd, 24, 30
• Solids of
Revolution
Worksheet