Geo 9 Ch 12
1
h r l
Volume and surface area Polyhedra
Solid Edges and Vertices

Geo 9 Ch 12
12-1 Prisms
2
We will start looking at three-dimensional solids and calculating various areas and volumes.
First, some definitions:
Prism – _________________________________________________________________
________________________________________________________________
Length of altitude _____________________________________________________
Lateral Faces ________________________________________________________
Lateral edges ________________________________________________________
Right Prism __________________________________________________________
Oblique Prism ________________________________________________________
Classify the following as accurately as possible.
__________________ _________________________ _________________
A = _____________ ________________ _____________
LA = _____________
TA = _____________
V = _____________
Memorize this immediately!!
________________
________________
________________
_____________
_____________
_____________
Platonic Solids rotate http://matti.usu.edu/nlvm/nav/frames_asid_128_g_4_t_3.html?open=instructions
Geoboard for 3-D Shapes http://matti.usu.edu/nlvm/nav/frames_asid_129_g_4_t_3.html?open=activities
* Prisms, Pyramids,etc http://www.mathsnet.net/geometry/solid/index.html http://www.pinkmonkey.com/studyguides/subjects/geometry/chap8/g0808101.asp

Geo 9 Ch 12
A. A cube has volume 8 cm 3 . Complete each statement.
1. Length of each edge _______ 1b. The total area of the cube is ___________
3
B. Use the figure to find the following:
2. A = _________________
3. P = _________________
4. LA = ________________
5. TA = ________________
6. V = _________________
Use the figure to find the following:
7. A = _____________________
8. P = _____________________
9. LA = _____________________
10.TA = _____________________
11. V = _____________________
5
6
3
12
4
10

13.
Geo 9 Ch 12
12.
10
8
10
20
6
10
10
Find the LA, TA, V
4

Geo 9 Ch 12 5
14. A drinking trough for horses is a right trapezoidal prism with the dimensions labeled.
If it is filled with water, about how much water will be needed to fill it two-thirds full?
100 cm
2m
50 cm
40 cm
15. Find the weight, to the nearest kilogram, of the cement block. Cement weighs 7860 kg per cubic meter.
12
10
20
40
20

Geo 9 Ch 12
16.
Find the weight, to the nearest 10 kg, of the steel
I -beam with the stated dimensions. Steel weighs
7860 kg per cubic meter.
5
20
5
30
5
6
10

Geo 9 Ch 12
12-2 Pyramids
Each pyramid has a vertex, a base and an altitude. Identify the following in the picture to the right.
Each pyramid has _____________faces with the ________ in common. These are the
7
__________________. They intersect in segments called ___________________.
Each pyramid is named by is base (regular hexagonal pyramid, square pyramid etc etc)
Most pyramids we study will be regular pyramids. They have the following properties:
1. __________________________________________________
2. __________________________________________________
3. __________________________________________________
4. __________________________________________________
To find the ____________ of a regular pyramid with _____ lateral faces you could find the area of
_____________________ and multiply it by _____________ or use theorem 12-3.
Theorem 12-3: The lateral area of a regular pyramid equals __________________________
____________ times ______________________. That is LA = _________.
Theorem 12-4: The volume of a regular pyramid equals _____________________________
_____________ times ____________________. That is V = ____________.
Total Area = ______________________________________

6. BA
8. TA
Geo 9 Ch 12 8
A regular triangular pyramid has slant height 9 and base perimeter 12. Find the following:
1. LA
2. BA h
3. TA
A regular square pyramid has base edge 6 m and lateral edge 5 m.
Complete:
4. Length of a slant height 5. LA
7. Length of altitude
9. V l k

Geo 9 Ch 12
Given the drawing, fill in the chart: Each row is a separate problem. k Height, h
Slant height, l
Base edge
Lateral edge
LA
3 4
5 13 h k l
TA
9
V

Geo 9 Ch 12 10
12-3 Cylinders and Cones
We will be dealing with “right” cylinders and cones, although oblique do exist.
Cylinder _______________________________________________________
Altitude _______________________________________________________
Radius _______________________________________________________
Theorem 12-5: The lateral area of a cylinder equals ______________________________
times _______________________. That is, LA = __________________.
Theorem 12-6: The volume area of a cylinder equals ______________________________
times _______________________. That is, V = __________________.
Theorem 12-7: The lateral area of a cone equals ______________________________
times _______________________. That is, LA = __________________.
Theorem 12-8: The lateral area of a cone equals ______________________________
times _______________________. That is, LA = __________________.

Geo 9 Ch 12
Fill in the chart, given the diagrams.
Cylinder
11
Cone
7
B
LA
5
13
TA
V
24
2. A cone with radius 6 cm and height 12 cm is filled to capacity with Cherry Coke

. Find the minimum height of a cylinder with radius 4 cm that will hold the same amount of liquid.
3. A right cone and a right cylinder have equal base areas. The height of the cylinder is the four times the height of the cone. Compare their volumes.

5.
Geo 9 Ch 12
4. r h
5 10
3 7
12 r
5 h
12 l
8
2
17
LA
LA
TA
TA
V
192

V
6 2
12 h r h l r

Geo 9 Ch 12
5.2
6.
6.8
13
Water is pouring into a conical reservoir at the rate of 1.8 cubic meters per minute. Find to the nearest minute, the number of minutes it will take to fill the reservoir.

Geo 9 Ch 12
6. 12-4 Spheres
A sphere is ___________________________________________________________
Theorem 12-9: The area of a sphere equals _________________________________
14
That is, A = _______________.
Theorem 12-10: The volume of a sphere equals _______________________________
That is, V = ______________.
1. Complete the following table for spheres. Leave answers in terms of
 d r A V
12
7
64

3
12
 r
2. If the surface area of a sphere is 16

, find the diameter and the volume.
3. Find the area of the circle formed when a plane passes 9 cm from the center of a sphere
with radius 15 cm.
4. Mr. Trem made 2 wax candles, one in the shape of a sphere with radius 5 cm and one in
the shape of a cylinder with radius 5 cm and height 6 cm. Which candle required more
wax?

Geo 9 Ch 12
4
12
15
20
8
3
6

Geo 9 Ch 12
H
1

4 , H
2

12
6
16

Geo 9 Ch 12
17
(1) A right prism has isosceles trapezoids for bases. The parallel edges of each base measure
6 and 12, while the two nonparallel edges measure 5. The height of the prism is 8. Find the;
(a) LSA , (b) TSA , and (c) volume of the prism.
(2) The radius of a sphere is 3. Find its volume and total surface area.
(3) A right cone is inscribed in a right square pyramid so that they have the same vertex, and the base of the cone is inscribed in the base of the pyramid. The area of the base of the pyramid is 144, and the common altitude is 8. Find the volume and total surface area of each.
(4) A sphere has a radius of 6. A right cylinder, having the same radius, has the same volume.
Find the height and TSA of the cylinder.
(5) A sphere whose radius is 3 is inscribed inside a right cylinder so that the sphere is tangent to both bases and the sides of the cylinder. Find the volume between the cylinder and sphere.
(6) A right pyramid has a regular hexagon for a base. Each edge of the base is 6, and each lateral edge measures 2 21 .
Find the; (a) LSA , (b) TSA , and (c) volume of the pyramid.
(7) A solid metal right cylinder is melted and re-cast as a sphere. The radius of the cylinder was
4 and the height was 18. Find the volume and surface area of the sphere.
(8) The radius of the base of a right cone is 6 and the height is 9. A cylindrical hole of radius 2 is drilled through the center of the base of the cone. Find the volume of the solid, which remains.
(9) A right cone is inscribed in a cube so that they have the same height, and the base of the cone is inscribed inside the base of the cube. The height of the cone is 6. Find the volume between the cube and cone.
(10) A cone is inscribed inside a right cylinder so that they have the same base, and the vertex of the cone is at the center of the top base of the cylinder. The height of the cylinder is 12, and the radius of the base is 5. Find the total surface area of the cylinder and cone.
(11) The base of a regular pyramid is a square. Each edge of the base is 16, and the height of the pyramid is 6. A plane, parallel to the base passes through the pyramid 3 units from its vertex. Find the volume and total surface area of the portion of the pyramid, which lies below the plane.
(12) Find the LSA , TSA , and volume of a cube if the length of one diagonal of the cube is 4 3 .

Geo 9 Ch 12
Answers
(1) (a) L.S.A. = 224 (b) T.S.A. = 296
(2) Volume = 36

, T.S.A. = 36

(c) Volume = 288
(3) V pyr
= 384 , T .
S .
A .
pyr
= 384 , V cone
= 96

, T .
S .
A .
cone
= 96

18
(4) height = 8 , T .
S .
A .
cyl
= 168

(5) V cyl
= 54

, V sphere
= 36

, V between
= 18

(6) (a) L.S.A. = 90 3 (b) T.S.A. = 144 3
(7) V sphere
= 288

, T .
S .
A .
sphere
= 144

(8) V rem
= 80

(9) V between
= 216

18

(10) T .
S .
A .
cylinder
= 170

, T .
S .
A .
cone
= 90

(11) Volume = 448 , T.S.A. = 560
(12) L.S.A. = 64 , T.S.A. = 96 , Volume = 64
(c) Volume = 216

Geo 9 Ch 12 19
(1) A right prism has isosceles trapezoids for bases.
The parallel edges of each base measure 6 and
12, while the two nonparallel edges measure 5.
The height of the prism is 8. Find the; (a) LSA ,
(b) TSA , and (c) volume of the prism.
6
5 5
12
8
5
6
5
8
12
(3) A right cone is inscribed in a right square pyramid so that they have the same vertex, and the base of the cone is inscribed in the base of the pyramid. The area of the base of the pyramid is 144, and the common altitude is 8. Find the volume and total surface area of each.
8
(2)
The radius of a sphere is 3. Find its volume and total surface area.
(4) A sphere has a radius of 6. A right cylinder, having the same radius, has the same volume. Find the height and TSA of the cylinder.
●
6
●
3
6

18
Geo 9 Ch 12
(5)
A sphere whose radius is 3 is inscribed inside a right cylinder so that the sphere is tangent to both bases and the sides of the cylinder. Find the volume between the cylinder and sphere.
(6)
3
●
(7)
A solid metal right cylinder is melted and re-cast as a sphere. The radius of the cylinder was 4 and the height was 18. Find the volume and surface area of the sphere.
(8) lateral edge measures 2 21 .
Find the; (a)
LSA , (b) TSA , and (c) volume of the pyramid.
20
A right pyramid has a regular hexagon for a base. Each edge of the base is 6, and each
2 21
2 21
2
6
The radius of the base of a right cone is 6 and the height is 9. A cylindrical hole of radius 2 is drilled through the center of the base of the cone. Find the volume of the remaining solid.
4
9 6

Geo 9 Ch 12
(9)
A right cone is inscribed in a cube so that they have the same height, and the base of the cone is inscribed inside the base of the cube. The height of the cone is 6. Find the volume between the cube and cone.
6
6
(11) The base of a regular pyramid is a square.
Each edge of the base is 16, and the height of the pyramid is 6. A plane, parallel to the base passes through the pyramid 3 units from its vertex. Find the volume and total surface area of the portion of the pyramid, which lies below the plane.
3
16
(10)
(12)
4 3 .
21
A cone is inscribed inside a right cylinder so that they have the same base, and the vertex of the cone is at the center of the top base of the cylinder. The height of the cylinder is 12, and the radius of the base is
5. Find the total surface area of the cylinder and cone.
12
5
Find the LSA , TSA , and volume of a cube if the length of one diagonal of the cube is
4 3

Geo 9 Ch 12
(1) (a)
(2)
(3)
(4) height = 8 ,
(5)
(7)
(8)
(9)
(11)
(12)
L.S.A. = 224
Volume = 36
V pyr
= 384 , T .
S .
A .
pyr
(6) (a) L.S.A. =
V sphere
90 3
V rem
= 80

V between

T .
S .
A .
cyl
V cyl
= 54

,
(10)
, T.S.A. = 36
V sphere
= 288

,
= 168

= 36

,
T .
S .
A .
sphere
= 216

18

T .
S .
A .
cylinder

(b)
Volume = 448 , T.S.A. = 560
Answers
T.S.A. = 296
= 384 , V cone
= 96

, T .
S .
A .
cone
V between
(b) T.S.A. =
= 144

= 170

, T .
S .
A .
cone
L.S.A. = 64 , T.S.A. = 96 , Volume = 64
= 18

144
= 90

3
= 96

22
(c) Volume = 288
(c) Volume = 216

Geo 9 Ch 12
CH 12 SUPPLEMENTARY PROBLEMS
23
Before 12.1
1) In the middle of the nineteenth century, hexagonal barns and sheds (and even some houses) became popular. How many cubic feet of grain would a hexagonal barn hold if it were 12 feet tall and had a regular base with 10 foot edges.
2) In some old buildings there were triangular pillars with the edges of the triangle of 10, 10 and
12 inches. The height of the pillar is 4 feet. How much granite would be needed to build these pillars?
Before 12.2
3) The base of the smaller Incan pyramid at Machu Picchu is a regular hexagon, which has sides of 20 meters. All six of the pyramid’s lateral edges are 30 meters long. Calculate the area of the six lateral faces and the base . What would the volume of the pyramid be? Use your books.
Before 12.3
4) The highway department keeps its sand in a conical storage building that is 64 feet in diameter and 24 feet high. To estimate the cost of painting the building, the lateral surface area of the cone is needed. To the nearest square foot, what is that area? Use 3.14 as

.
5) A conical ice cream cone is 2” in diameter and 5” deep. How much melted ice cream will it hold?
Before 12.4
6) A 10 cm tall cylindrical glass 8 cm in diameter is filled to 1 cm from the top with water. If a gold ball 4 cm in diameter is dropped into the glass, will the water overflow?
Before 12.5
7) The volumes of two similar hexagonal prisms are in the ratio of 8:125. What is the ratio of their heights?
8) The surface areas of two cubes are in the ratio of 49:81. If the volume of the smaller cube is
20, what is the volume of the larger cube?