Transcript Geometry - BakerMath.org

Geometry
Surface Area and Volume of Spheres
Goals
Find the surface area of spheres.
 Find the volume of spheres.
 Solve problems using area and volume.

July 28, 2015
Sphere Demo
Sphere
The set of points in
space that are
equidistant from the
same point, the center.
radius
Great Circle
(Divides the sphere
into two halves.)
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Hemisphere
Half of a sphere.
r
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Sphere Formulas
SA  4 r
r
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V  r
4
3
3
2
Using a Calculator

You may find it easier
to use the formula for
volume in this form:
4

r
V
3
July 28, 2015
3
SA  4 r
Example
Find the Surface Area and
the Volume of a sphere
with a radius of 2.
2
 
 4 2
 16
 50.3
 
 2 
V  r
4
3
2

4
3

32
3
3
3

 33.5
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2
Your Turn
Find the surface area and volume.
7 in.
 
SA  4 7
2
 196
 615.8
 
V  7
4
3
3
1372


3
 1436.8
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Problem 1

A snowman is made with three spheres.
The largest has a diameter of 24 inches,
the next largest has a diameter of 20
inches, and the smallest has a diameter of
16 inches. Find the volume of the
snowman.
July 28, 2015
Problem 1 Solution


A snowman is made with three spheres. The largest has
a diameter of 24 inches, the next largest has a diameter
of 20 inches, and the smallest has a diameter of 16
inches. Find the volume of the snowman.
The radii are 12 in, 10 in, and 8 in.
July 28, 2015
Problem 1 Solution

The radii are 12 in, 10 in, and 8 in.
 
 10   4188.79
 12   7238.23
V   8  2144.66
4
3
V
4
3
V
4
3
3
3
3
13,571.68 cu in
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Or, for easier calculation…
V   (8 )   (10 )   (12 )
4
3
3
3
4
3
  (8  10  12 )
4
3
3
  (3240)
4
3
 13,571.68
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3
3
4
3
3
Problem 2
This is a grain silo, as found on
many farms. They are used to
store feed grain and other
materials. They are usually
cylindrical with a hemispherical
top.
Assume that the concrete part has
a height of 50 feet, and the
diameter of the cylinder is 18 feet.
Find the volume of the silo.
July 28, 2015
Problem 2 Solution
18
Volume of Cylinder
V = r2h
50
9
V = (92)(50)
V =   81  50
V = 4050
V  12723.5 cu. ft.
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Problem 2 Solution
18
Volume of Hemisphere
50
9
 
V  9
4
3
3
 972
 3053.6 cu. ft.
This is the volume of a
sphere. The volume of the
hemisphere is half of this
value, which is 1526.8 cu. ft.
July 28, 2015
Problem 2 Solution
18
Volume of Cylinder
12723.5
50
9
Volume of Hemisphere
1526.8
Total Volume
12723.5 + 1526.8 =
14250.3 cu. ft.
July 28, 2015
Problem 2 Extension
Total Volume =14250.3 cu. ft.
One bushel contains 1.244
cubic feet. How many
bushels are in the silo?
14250.3  1.244 =
11455.2 bushels
July 28, 2015
Skip
Problem 3
July 28, 2015
A sphere is inscribed
inside a cube which
measures 6 in. on a side.
What is the ratio of the
volume of the sphere to
the volume of the cube?
Problem 3 Solution
Volume of the Cube:
6  6  6 = 216 cu in
6
Radius of the Sphere:
6
3
6
3 in.
Volume of the Sphere:
 
V  3
4
3
3
 43  (27)
 36
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Problem 3 Solution
Volume of the Cube:
216 cu in
Volume of the Sphere:
6
36
6 Ratio of Volume of Sphere
3
to Volume of Cube
6
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36
36



216 36  6 6
Problem 4
A mad scientist makes a potion
in a full spherical flask which has a
diameter of 4 inches. To drink it, he
pours it into a cylindrical cup with a
diameter of 3.5 inches and is 3.5
inches high. Will the potion fit into the
cup? If not, how much is left in the
flask?
Skip
July 28, 2015
Problem 4 Solution
Flask Volume:
Diameter = 4 inches
Radius = 2 inches
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 
V  2
4
3
3
4 8

3
 33.5 cu in
Problem 4 Solution
33.5 cu in
Cup Volume:
Diameter = 3.5 inches
Radius = 1.75 inches
Height = 3.5 inches
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V   (1.75 )(3.5)
2
 33.7 cu in
Problem 4 Solution
33.5 cu in
33.7 cu in
The flask holds 33.5 cu in.
The cup holds 33.7 cu in.
Yes, the potion fits into the cup.
July 28, 2015
Problem 5
Skip
The surface area of a sphere is 300 cm2.
Find:
1. Its radius,
2. The circumference of a great circle,
3. Its volume.
July 28, 2015
Problem 5 Solutions
SA  4 r
2
300  4 r
2
300
r 
 23.87
4
2
C  2 r
 2 (4.89)
 30.72
V  r
4
3
3
r  23.87
  (4.89 )
r  4 .8 9
 489.8
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4
3
3
Summary
A sphere is the set of points in space
equidistant from a center.
 A hemisphere is half of a sphere.
 A great circle is the largest circle that can
be drawn on a sphere. The diameter of a
great circle equals the diameter of the
sphere.

July 28, 2015
Formulas to Know
SA  4 r
r
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V  r
4
3
3
2
Skip
Last Problem
On a far off planet, Zenu
was examining his next
target, Earth. The radius of
the Earth is 3963 miles.
What is the volume of
material that will be blown
into space?
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Last Problem Solution

V  43  39633

 2.607 10 cubic miles
 260,711,883,000 cu mi
11
July 28, 2015
Practice Problems
July 28, 2015