Volume-Lateral Area-Total Area
Right
Circular
page #10
Cylinders
A right circular cylinder is like a right prism except that its bases are congruent circles
instead of congruent polygons.
base
height
radius
base
Lateral Area of a Right Circular Cylinder:
Imagine the diagram shows a can with its ends removed. If you cut along AB, you can
unroll the metal and lay it out flat.
B
B
h
A
h
A
The lateral area becomes
the area of this rectangle:
L.A. = π dh or
L.A. = 2 π r h
Circumference of the base
C = πd
Total Area of a Right Circular Cylinder:
The total area of a right circular cylinder is the lateral area plus the area of the
two bases:
T.A. = π dh + 2 π r 2 or T.A. = 2 π rh + 2 π r 2
Volume of a Right Circular Cylinder:
The volume of a right circular cylinder is the area of one base times the height.
V = Bh
V = π r 2h
Volume-Lateral Area-Total Area
page #11
Example 1: Find the lateral area, total area and volume of the following right circular
cylinder:
L.A. = 2 π rh = 2 π (3)(4) = 24 π cm2
4 cm
T.A. = 2 π rh + 2 π r 2
T.A. = 24 π + 2 π (3)2 = 24 π + 18 π = 42 π cm2
3 cm
V = π r 2h = π (3)2(4) = 36 π cm3
Example 2: Find the radius of the cylinder if its height is 10 meters and its volume is
90 π square meters.
V = Bh
V = π r 2h
90 π = π r 2(10)
90π 10πr 2
=
10π
10π
9 = r2
3 meters = r
Problems:
1. Find the lateral area, total area and volume of the following right circular cylinder:
2
3
Volume-Lateral Area-Total Area
page #12
2. Find the radius of a cylinder if its height is 6 feet and its volume is 36 π feet3.
3. Find the height of a cylinder if its radius is 4 feet and its volume is 48 π ft3.
4. The total area of a cylinder is 40 π . If h = 8, find r.
Volume-Lateral Area-Total Area
page #13
Right Circular Cone
In the right circular cone, label the base, radius, height of the cone and slant height.
Lateral Area, Total Area and Volume of a Right Circular Cone
Lateral Area:

h
Total Area: Lateral Area + B
Total area: π r  + π r 2
1
Bh
3
1
V = π r 2h
3
V=
r
πr 
Volume-Lateral Area-Total Area
page #14
Example 1: Find the lateral area, total area and volume of the cone:
r 2 + 82 = 10 2
10
8
r 2 + 64 = 100
r 2 = 36
r=6
r
Lateral Area:
π (6)(10) = 60 π
Total Area: 60 π + π (6)2 = 60 π + 36 π = 96 π
Volume:
1
1
1
π (6)2(8) = π (36)(8) = π (288) = 96 π
3
3
3
Example 2: The volume of a right circular cone is 48 π cubic units and its altitude is 4
units. Find the radius of the base.
1
V = πr 2 h
3
1
48π = πr 2 (4)
3
144π = πr 2 (4)
144 = 4r 2
36 = r 2
6=r
Volume-Lateral Area-Total Area
page #15
Problems:
1. Find the lateral area, total area and volume of the cone.
13
12
r
2. Find the height of a right cone whose volume is 924 π cm3, and whose base has a
radius of 14 cm.
3. Find the radius of a cone with lateral area of 72 π and slant height of 9.
Volume-Lateral Area-Total Area
page #16
4. The radius of the base of a cone measures 3 units and the altitude measures 7
units. Find the volume.
5. The volume of a cone is 320 π cm3 and the altitude measures 15 cm. Find the length
of the radius of the base.
6. Water is pouring into a conical reservoir at the rate of 1.8 m3 per minute. How long
will it take to fill the reservoir?
5.2 m
6.8 m
Volume-Lateral Area-Total Area
Spheres
Area = 4πr 2
r
Volume =
4 3
πr
3
Example 1: The diameter of a sphere is 8. Find the area and volume.
Since d = 8, r = 4.
Area = 4 π r 2
Volume =
4
πr3
3
Area = 4 π (4)2
Volume =
4
π (4)3
3
Area = 4 π (16)
Volume =
4
π (64)
3
Area = 64 π
Volume =
256π
3
Example 2: The area of a sphere is 256 π . Find the volume.
4 π r 2 = 256 π
r2 = 64
r=8
V=
4
πr3
3
V=
4
π (8)3
3
V=
4
π (512)
3
V=
2048π
3
page #17
Volume-Lateral Area-Total Area
Problems:
1. Find the area and volume of the sphere.
9
2. Find the area and volume of the sphere.
10
3. A baseball has a radius 7 cm long. Find its surface area.
4. Find the volume of a sphere with a 3 cm radius.
page #18
Volume-Lateral Area-Total Area
page #19
5. Find the length of a radius of a sphere whose area is 196 π cm .
2
6. How many cm3 of air can be pumped into a basketball if its maximum diameter is 20
cm?
7. Find the length of a radius of a sphere whose volume is 972 π cm3.
8. The area of a sphere is 400 π . Find the volume.
Volume-Lateral Area-Total Area
page #20
Formulas
B = the area of one base
1.
Right Prism
L.A. = sum of the areas of the lateral faces
T.A. = L.A. + 2B
h
V = Bh
h
2.
Right Circular Cylinder
L.A. = 2 π r h
T.A. = L.A. + 2B = 2 π rh + 2 π r 2
h
V = Bh = π r 2h
r
3.
Regular Pyramid
L.A. = sum of the areas of the lateral faces
T.A. = L.A. + B

h
V=
4.
1
Bh
3
Right Circular Cone
L.A. = π r 

h
r
5.
T.A. = L.A. + B = π r  + π r 2
V=
1
1
Bh = π r 2h
3
3
Sphere
A = 4πr 2
r
V=
4 3
πr
3
Volume-Lateral Area-Total Area
page #21
Review Volume, Lateral Area, Total Area
1. Find the lateral area, total area and volume:
15 m
10 m
9m
12 m
2. The walls and ceiling of a warehouse are to be painted. How many square meters
must be covered if the warehouses is 120 meters by 96 meters with a 3 meter high
ceiling?
3. Find the lateral area, total area and volume:
12
18
15
Volume-Lateral Area-Total Area
page #22
4. A side of the base of a square pyramid measures 6 units and a lateral edge
measures 5 units. Find the total area.
5. Find the lateral area, total area, and volume:
5
8
6. Find the radius of a right circular cylinder with lateral area of 216 π and height of
12.
Volume-Lateral Area-Total Area
page #23
7. Find the lateral area, total area and volume:
26
24
r
8. What is the slant height of a cone with lateral area of
9. Find the area and volume:
4
9π
1
and radius of ?
10
5
Volume-Lateral Area-Total Area
page #24
10. The area of a sphere is 484 π . Find the volume of the sphere.
11. A pyramid has a rectangular base 10 cm long and 6 cm wide. The pyramid’s height
is 4 cm. Find the volume.
12. A cone has a radius 8 and height 6. Find the lateral area, total area and volume.
Volume-Lateral Area-Total Area
page #25
13. The total area of a cylinder is 18 π . If h = 8, find r.
14. The volume of a cylinder is 72 π . If h = 8, find L.A.
15. The area of the base of a cone is 49 π square units and the slant height measures
20 units. Find the length of the altitude.