3. MALAYAN COLLEGES LAGUNA
CONE
A cone is a solid bounded by a conical surface (lateral surface) whose directrix i
s a closed curve, and a plane (base) which cuts all the elements.
Properties
1. The altitude of a cone is the perpendicular
distance from the vertex to the plane of the base.
2. Every section of a cone made by a plane passing
through its vertex & containing two points of
base is a triangle.
3. The axis of a cone is the straight line joining
vertex with the center of the base.
4. A right section of a cone is a section perpen-
dicular to its axis & cutting all the elements.
5. A circular cone is a cone whose right section
is a circle
3
vertex
e
B
h
5. MALAYAN COLLEGES LAGUNA
CONE
A right circular cone is a circular cone whose axis is perpendicular to its base.
Properties
1. The slant height of the right circular
cone is the length of an element.
2. The altitude of a right circular cone is the
distance between the vertex & the center
of the circle which forms its base.
3. A right circular cone is a solid generated
by rotating a right triangle about one of
its legs as an axis.
4. All elements of a right circular cone are equal.
5. A section of a right circular cone parallel to the base
is a circle whose center is on the axis of the cone.
6. A section of a right circular cone which contains the
vertex & two points of the base is an isosceles triangle.
5
h
l
r
6. MALAYAN COLLEGES LAGUNA
CONE
Formula:
The volume of the cone is equal to one-third the pr
oduct of the base and the altitude.
Volume = 1/3 base x altitude
V = 1/3 Bh
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7. MALAYAN COLLEGES LAGUNA
CONE
Formulas:
The volume of a right circular cone is equal to one-thir
d the product of the base and the altitude.
Volume = 1/3 base x altitude
V = 1/3 Bh
The lateral area of a right circular cone is equal to on
e-half the product of the circumference of the base a
nd the slant height.
Lateral Area = ½ circumference of base x slant height
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11. MALAYAN COLLEGES LAGUNA
CONE
PROBLEMS
Two vertical conical tanks of the same diameter which is equal to 6 feet a
re connected by a pipe line 3 inches in diameter and 24 feet in length. Th
e bigger tank with an altitude equal to 10 feet is initially full of water whi
le the smaller tank of altitude 6 feet is empty. If the pipe valve is open to
allow water to flow from the bigger tank to the smaller tank untl it is full,
find the height of the water remaining in the tank.
The crater of a volcano is approximately in the shape of a cone of base 3.
1416 sq. mile. The crater’s depth is 1500 ft. How many cubic yards of ea
rth will be required to fill this cavity? ( Ans: 16.219 x 108 cu. yard )
A pile of sand is in the form of a right circular cone of altitude 7 ft. and sl
ant height 25 ft. What is the weight of the sand, if the sand weighs 107.5
lbs / cu. ft.? ( Ans: 453,900 lbs. )
How many square yards of canvas will be required to make a conical tent
15 ft. high and 18 ft. in diameter, if 10% of the material is wasted? ( An
s: 61.061 sq. yd. )
A right circular cone of slant height 10 in. has a radius of 4 in. Find the a
ngle of the sector of a circle of radius 10 in. whose area is equal to the lat
eral area of the cone. ( Ans: 144o
) 11
12. MALAYAN COLLEGES LAGUNA
CONE
PROBLEMS
Find the volume and the total surface area of a right circular cone whose
base radius is 8 cm. and whose altitude is 15 cm. (Ans: 320 cu. cm.; 20
0 sq. cm. )
A cylindrical tower 30 ft. in diameter has a conical roof the length of who
se eves is 2 ft. An element of the roof is inclined 450
to the horizontal. Fin
d the weather surface. ( Ans: 1197 sq. ft.)
A piece of lead pipe of inner diameter 2 ¼ in., outer diameter of 2 5/8 i
n., and length 16 ft. has been melted in an open conical pot of radius 10 i
n and altitude 15 in. Find the depth of the molten metal? ( Ans: 8.40 in.
)
12
13. MALAYAN COLLEGES LAGUNA
CONE
PROBLEMS
Find the least waste in cutting two conical blocks from a block of wood in
the form of a right circular cylinder of radius 4 in. and altitude 7 in. ( An
s: 117.29 cu. in.)
A right circular cone has an altitude of 18 cm. and radius of the base is 5
cm. Inscribe in it is a square pyramid. Find the volume of the cone and th
e pyramid. ( Ans: 150 cu.cm. ; 300 cu. cm.)
The ratio of radii of cone and cylinder is 1:2 while the ratio of radius of co
ne to its altitude is 1:3. If lateral surface area of the cylinder equals the v
olume of cone, find the radius of the cone if the altitude of the cylinder i
s 4 units. (Ans: 4 units)
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