Finding the surface area and volume of solids. Download the power point presentation to enable animation. Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.

1. PRISMS

2. PRISMS
A three dimensional geometric
figure with two identical
polygonal bases that aligned
in parallel and with
parallelogram on each side.

3. Triangular Prism

4. Rectangular Prism

5. Hexagonal Prism

6. SURFACE AREA OF
PRISMS
 When we want to find the surface area
of prisms, we can unfold it so that it
becomes many connecting two-
dimensional geometric figures and
then we calculate the sum of those
areas of those figures to obtain the
surface area of the original prism.

7. unfold
Right Triangular Prism
c
a b
h
a
ba
b
c
c
b
ba
a
h
1 2 3
4
5

8. a
h
a
a
a
a
a
a a a
a
a
h
Square Prism
1 2 3 4
5
6

9. a
b
h
a
b
a
b
a
a b
a
Rectangular Prism
1 2 3 4
5
6

10. a
a
h
a
a a
a
a
h
Hexagonal Prism
1
2
3
4
5
6
7
8

11. Examples:
1. Find the surface area of a square prism
given that each side of the base is 5 m and
the height of the prism is 12 m.
5m
12 m

12. Examples:
2. Find the surface area of a right
triangular prism that the length of a leg of
the right triangle is 5 inches and the
length of the other leg is 12 inches while
the height of the prism is 16 inches.
12 in 5 in
16 in

13. Examples:
3. Find the surface area of a regular
hexagonal prism of that the length of each
side of the hexagonal base is 2
centimeters and the height is 8
centimeters.
2 cm
8 cm

14. 4. Find the surface area of the triangular
prism below:
Examples:

15. Exercises:
1. Find the surface area of a right
triangular prism that the length of a leg of
the right triangle is 5 inches and the
length of the other leg is 12 inches while
the height of the prism is 16 inches.
2. Find the surface area of a square prism
given that each side of the base is 4 m and
the height of the prism is 11 m.

16. TRY THIS
1. A house with an equilateral triangular roof and
rectangular walls is to be painted with red. What is
the surface area of the house if its exterior is to be
painted? (Refer to the diagram below)
13 m
9 m
16 m
13 m

17. 2. A pipe with a cross-shaped base is meant to
connect a water source and a house. What is the
exterior surface area of the pipe? (Refer to the
diagram below)
TRY THIS
4 cm
6 cm
6 cm
2600 cm
4 cm
4 cm
6 cm
6 cm
4 cm

18. VOLUME OF
PRISMS
 the amount of space that a substance
or object occupies.
 volumes of prisms can be obtained
by multiplying the area of the base to
its height.

19. Rectangular Prism
a
b
h
V=Ah
V=abh
V is the volume
A is the base area
h is the height

20. Right Triangular Prism
a
b
h
V=Ah
Rectangular prisms
Right Triangular prisms
V=
1
2
Ah
V=
1
2
abh

21. Hexagonal Prism
Hexagonal prisms
h
a
Base Area
A =
3 3
2
𝑎2
Volume
V=Ah
V=
3 3
2
𝑎2ℎ

22. Examples:
1. Find the volume of a trapezoidal prism
of which the length of the parallel lines of
the bases are 5 and 8 inches and the
height of the trapezoid is 4 inches. The
height of the prism is 12 inches.
5 in
8 in
4 in
12 in

30. 180

32. Stephanie just redecorated her bedroom. She wants to paint her
door green.
200 cm
80
cm
3 cm
ANSWER: 33,680 cm2

33. Jordan built her cat Tuna a new scratching post. She needs to
cover the post with carpet. How much carpet does Jordan need
to cover the surface of the post, including the bottom?
10 cm
90 cm
10 cm
ANSWER: 3,800 cm2

34. Nemo's aquarium is filled with 2400 cubic centimeters of water.
The base of the aquarium is 20 centimeters long and 12
centimeters wide. What is the height of the water in Nemo’s
aquarium?
ANSWER: 10 cm
20 cm
12 cm
ℎ

35. Erin built a rectangular trough for her horses to eat from. The
trough is 3 meters long, 2 meters wide, and 2 meters high. Hay
for the trough costs $14 per cubic meter. How much will it cost
to completely fill the trough with hay?
3 m
2 m
2 m
ANSWER: $168

36. An exotic fish is placed in a rectangular aquarium that has a
length of 75 cm ,width of 35 cm, and a height of 24 cm. If the
water level rises 2 cm when the fish is placed, what is the volume
of the fish?
75 cm
35 cm
2 cm
24 cm
ANSWER: 5,250 cu. ft

37. A shipping company wants to ship its boxes in a 8-feet wide, 20-feet long, and
9-feet high trailer truck. The boxes have dimensions of 2 feet by 5 feet by 3
feet (2ft × 5ft × 3ft). What is the total volume of boxes that can fit in the trailer
if they have to be stacked so that the bottom of each box measures 3 feet
wide and 5 feet long and two boxes are placed in each row?
8 ft
20 ft
5 ft
3 ft
9 ft
2 ft
ANSWER: 960 cu. ft

38. Cylinder
Cross section or
base
axis
height
Right cylinder is a three-dimensional geometric figure with parallel cross
sections or bases. Each bases of the right cylinder is vertical to its axis.
Oblique cylinder is a three-dimensional geometric figure with parallel cross
sections or bases. Each bases of the oblique cylinder is, however, not vertical
to its axis.

39. Surface Areas of Cylinders
We can find the surface area of a cylinder by unfolding the
cylinder into two-dimensional geometric figures and find the
total surface areas of the figures.
h
r

40. Example 1
Find the surface area of a cylinder whose base
diameter is 14 centimeters and the height is 4
centimeters. 𝜋 ≈
22
7
4 cm
14 cm

41. Example 2
Find the surface area of a three-dimensional
geometric figure that each base of this figure is half of
a 3.5 centimeter radius circle and the length of it is 12
centimeters long as shown in the figure below. 𝜋 ≈
3.5 cm
12 cm

42. Example 3
A cylindrical trash can is 4 feet high with 8 feet diameter.
If the side, not the top nor the bottom, of the can has to
be painted with the paint that costs 30 baht per square
foot, how much money is needed? 𝜋 ≈
22
7
8 ft
4 ft

43. Find the outer surface area 𝜋 ≈
22
7

44. Find the inner surface area 𝜋 ≈ 3.14
interior

45. Find the inner surface area 𝜋 =
22
7
interior

46. Volume of Cylinders
pentagon hexagon octagon
nonagon decagon

47. A polygon with an infinite number of
sides is called a circle.

48. Volume of Prisms
Area of the base
heightV=Ah
A
h
Volume of Cylinders
V=Ah
A
h
= 𝜋𝑟2
V=𝜋𝑟2
h

49. Example 1
Find the volume of the cylinder whose diameter is 4
centimeters and height is 9 centimeters. 𝜋 ≈
22
7
9cm
4 cm

50. Example 2
Find the height of a cylinder whose diameter is 8
inches and volume is 120 cubic inches. 𝜋 ≈
22
7
h
8 inches

51. Example 3
Find the amount of plastic material used in making a
water pipe with the outer diameter of 2 centimeters,
the thickness of 2 millimeters, and the length of 25
centimeters. 𝜋 ≈
22
7
25 cm
2 cm
1 cm

52. 1. A company is deciding which box to use for
their merchandise. The first box measures 8
inches by 6 inches by 10 inches. The second box
measures 9 inches by 5 inches by 12 inches. If
each box costs 25 baht per square inch to make,
how much does the company save by choosing
the cheaper box if the company needs fifty
boxes?

53. 2. A cosmetics company that makes small
cylindrical bars of soap wraps the bars in plastic
prior to shipping. Find the surface area of a bar of
soap if the diameter is 14 centimeters and the
height is 12 cm. 𝜋 ≈
22
7

54. 3. A farmer needs to make a barn using the plan
below. How much wood does he need if he has
to cover the exterior of the barn?
13
13
10
18
16

55. A shipping company wants to ship its boxes in a 8-feet wide, 20-feet long, and
9-feet high trailer truck. The boxes have dimensions of 2 feet by 5 feet by 3
feet (2ft × 5ft × 3ft). What is the total volume of boxes that can fit in the trailer
if they have to be stacked so that the bottom of each box measures 3 feet
wide and 5 feet long and two boxes are placed in each row?
8 ft
20 ft
5 ft
3 ft
9 ft
2 ft
ANSWER: 960 cu. ft

56. 5. A cylindrical golden tube with a diameter of 21
inches and a height of 9 inches is to be melted
into right triangular golden prims, wherein the
legs of its triangular base are 8 inches and 15
inches and its lateral area is 240 square inches.
How many right triangular golden prisms can be
made from the cylindrical golden tube?

57. PYRAMI
D

58. Pre-test
A square pyramid has a base with the side length of 6
centimeters and the slant height of 5 centimeters. Find
the surface area of the pyramid.

59. 75°75°
75°
75°
75°
75°

61. Apex
Lateral Edge
Lateral Face
Height
Base
Right Pyramid Oblique Pyramid

62. Triangular
Pyramid
Rectangular
Pyramid
Square
Pyramid
Hexagonal
Pyramid

63. Surface Area
PYRAMID
S
of
To find the surface area of a pyramid, we can unfold it
and find the sum of the areas from two- dimensional
figures.

64. a
a
𝑙
a
𝑙
Slant height
Equilateral Triangular Pyramid

65. a
𝑙 Slant height
Square Pyramid
𝑙
a
a

66. 𝑙
a
a Slant height𝑙
a

67. Example 1
A square pyramid has a base with the side length of 6
centimeters and the slant height of 5 centimeters. Find
the surface area of the pyramid.

68. 5 cm
6 cm 5 cm
6 cm
Example 1
A square pyramid has a base with the side
length of 6 centimeters and the slant height
of 5 centimeters. Find the surface area of the
pyramid.

69. Example 2
The base of an equilateral triangular pyramid has the
side length of 8 centimeters. All of the lateral faces of
the pyramid, including the base, are identical. Find the
surface area of the pyramid.

70. Example 2
The base of an equilateral triangular pyramid has the side
length of 8 centimeters. All of the lateral faces of the
pyramid, including the base, are identical. Find the surface
area of the pyramid.
8 cm
8 cm
8 cm
8 cm

71. Example 3
A rectangular pyramid whose base is 4 centimeters wide
and 6 centimeters long and has the length of the lateral
edges of 8 centimeters. Find the surface area of the
pyramid.

72. Example 3
A rectangular pyramid whose base is 4 centimeters wide and 6
centimeters long and has the length of the lateral edges of 8
centimeters. Find the surface area of the pyramid.
6 cm
6 cm
4 cm
4 cm
8 cm
8 cm
𝑙1
𝑙2
8 cm

73. Volume of Pyramids

74. Volume of Pyramids
1
3
V = Ah_
base area
height

75. Examples:
1. Find the volume of a 10 centimeteres high
reactangular pyramid whose base is 6
centimeters wide and 8 centimeters long.
10 cm
8 cm
6cm

76. Examples:
2. A pyramid with a square base is 6 centimeters
high and has the volume of 32 cubic centimeters.
Find the a) base area of the pyramid. b) the
length of each side of the base.
6 cm
x
x
V = 32 cu. cm

80. CONES

81. Cones
A three-dimensional geometric figure with a circular base.
Apex
Slant height
Axis
Base
Height Height
Right Cone Oblique Cone
A right cone has its height which is vertical to the base through the center of the base. Its height
and its axis are the same line. All line segments connecting the apex to any points on the perimeter
of the base are equal in length and called slant heights.
An oblique cone has its height which is also vertical to the base but not at the center of the base.
Its height is not the same line as the axis

82. Surface Areas of Cones
r
h
𝑙
r
2𝜋r
𝑙
2𝜋r

83. _____________
__
2𝜋𝑟
𝑙
C
L
Area of L__________
Area of C
=
Arc length of L
Perimeter of C
_____________
__
Area of L__________
𝜋𝑙2
=
2𝜋𝑟
2𝜋𝑙
Area of L =
2𝜋𝑟 𝜋𝑙2
2𝜋𝑙
_____________
__
Area of L = 𝜋𝑟𝑙
Surface Areas of Cones

84. Area of L = 𝜋𝑟𝑙
𝑙
2𝜋r
r
2𝜋r
Area of B =
Surface Areas of Cones
𝜋𝑟2

85. Surface Areas of Cones
r
h
𝑙 Surface Area = Area of B +Area of L
= 𝜋𝑟2
+ 𝜋𝑟𝑙

86. Example 1
Find the surface area of a cone whose base radius is
7 centimeters and the slant height is 15 centimeters.
𝜋 ≈
22
7

87. Volume of Cones

88. Volume of Cones
1
3
V = Ah_
base area
height
1
3
V = 𝜋r2
h_

89. Example 2
Find the volume of a cone whose base radius is 3.5
centimeters and the height is 12 centimeters. 𝜋 ≈
22
7

90. Surface Areas of Spheres

91. Volumes of Spheres