Transcript 12.7 Prisms - Shelton State

A prism is a solid whose sides
(lateral sides) are parallelograms
and whose bases are a pair of
identical parallel polygons. A
polygon is a simple closed figure
whose sides are line segments.
Bases
Rectangular prism
Pentagonal prism
Triangular prism
The volume of a solid is the
number of cubes it takes to fill the
solid.
The volume of a prism is found by
multiplying the area of the base (B)
by the height of the prism. The
height is the distance between the
2 bases.
V  Bh
Find the volume of a rectangular
prism that has length of 7cm, with
of 6 cm and height of 4 cm.
4 cm
6 cm
7 cm
3
Steel weighs 0.28 lb / in . What is
the weight of a rectangular piece of
steel 0.25 in. by 15.0 in. by 32.0 in?
A cylinder is a geometric solid with
a curved lateral surface. A can is
an example of a cylinder.
The volume of a cylinder is given
by
r
V  Bh
h
 r h
2
Example: Find the volume of the
cylinder.
V  r h
2
d = 24 m
Since d = 24, then r = 12 m.
40 m
V  3.14(12) (40)
2
V  18086.4m
3
The volume of any cone or pyramid
is given by the formula
1
V  Bh
3
where B = area of the base
Apex
Slant height
height
height
diameter
Base
Base
Find the volume.
1
V  Bh
3
1
 (8.7  8.7)(6.5)
3
3
 193.995in
Find the volume.
19.6 cm
1
V  Bh
3
1
2
 (3.14 12 )(19.6)
3
3
 2954in
The volume of a sphere is given by
the formula
Vsphere
4r

3
3
43.148
V
3
3
 2143.6m
3
The lateral surface area is the sum
of the areas of the lateral faces of
the prism.
LSA = ph,
where p is the perimeter of the
base and h is the height of the
prism.
Find the lateral surface area of a
rectangular prism that has length of
7cm, with of 6 cm and height of 4 cm.
4 cm
6 cm
7 cm
Front and back = 4 x 7 each = 2(28) = 56 sq. cm.
2 ends = 4 x 6 each = 2(24) = 48 sq. cm.
Lateral surface area = 104 sq. cm.
The total surface area is found by
finding the sum of the lateral area
faces and the areas of the bases.
TSA = ph + 2B
4 cm
6 cm
7 cm
Lateral surface area = 104 sq. cm.
The top and bottom are the bases.
Top area = 6 x 7 = 42 sq. cm.
Same for the bottom = 42 sq. cm
Area of the bases = 2(42) = 84 sq. cm.
Total S.A. = 104 + 84
= 188 sq. cm.
The lateral surface area of a cylinder
and be visualized by taking a can,
cutting out the top and bottom, then
down the side and unrolling the can.
The resulting shape is a rectangle
that has length equal to the
circumference of the circular top and
width equal to the height of the can.
The formula is
L.S. A.  ph  2rh
The total surface area is the sum of
the lateral area and the 2 bases
(top and bottom)
•
Find the lateral surface area and
total surface area of the cylinder.
Lat. S.A.= 2rh
2.5 in
 2(3.14)(2.5)(12)
 188.4in
2rh
2
• Total S.A.= Lat.S.A. + 2 bases,
where the bases are circles
 188.4  2(3.14)(2.5) 2
 188.4  39.25
 227.65in
2
12 in
A steel cylindrical tank needs to hold
7000 gal. Due to space constraints, the
tank should be 10 ft in diameter. How tall
should the tank be? (Water weighs 8.34
lb/gal and 62.4 lb/cu.ft.)
• First convert gal to cu.ft.
 8.34lb  cu. ft. 

7000gal
  935.58cu. ft.
 gal  62.4lb 
• Take this volume and
radius of 5 ft, substitute
them into the volume
formula and solve for h.
Example continued:
V  r h
2
935.58  3.14(5) h
2
935.58  78.5h
11.92 ft  h
Find the amount of paper used for
labels for 1000 cans like those
shown below.
3.16 cm
Sweetheart
Chicken
Soup
8.24 cm
The total surface area of a sphere
is given by
TSA = 4πr²
TSA  4r
2
 43.148
2
 803.84m
2
Lateral surface area of a cone is
given by
LSA = πrs, where r is the radius
and s is the slant height,
and the total surface area is given
by
TSA = πrs + πr²
Find the lateral surface area and
total surface area of a cone that
has a radius of 6 ft, slant height of
10 ft and height of 8 ft.
LSA  rs
 3.14610
 188.4 ft
2
TSA  rs  r
2
 188.4  3.146 
 188.4  113.04
2
 301.44 ft
2