Maths

2. CONE
Q. What is circular cone ?
Answer :A circular cone is a surface generated by
a straight line passing through a fixed
point and moving on a circle.
Q. What is a right circular cone ?
Answer : Right circular cone: A right circular
cone is a surface generated by revolving
a straight line, which passes through a
fixed point and makes a constant angle
with a fixed line.
In all the above cases, hollow cone is generated.

3. TYPES OF CONES
Hereafter, we mean by cone a right circular cone. In
the figure, D is the vertex of a cone, the vertical
distance between the vertex and base of the cone is
called its height.

4. Height of the cone
The length of the segment is the height of the cone and
is usually denoted by h.
Slant height of the cone
The distance between the vertex and any point on the
circumference of the base circle is called its slant
height. The length of the segment is called the slant
height of the cone and is generally denoted by l.
Radius of the cone
The radius of the base circle is called the radius of the
cone and is usually denoted by r.
In a rt. angled triangle,
l x l =(h x h) + (r x r)
l= h + r2 2

5. DERIVATION
VOLUME OF A RIGHT CIRCULAR CONE
Volume of a cone
= 1/3 x π (r x r) h
The above formula can be verified
experimentally. Take a cylindrical
jar of height h and radius r whose
volume is π x (r x r) h.
Take a hollow cone which has the same
height h and same radius r. Fill the
cone with water and pour it into the
jar. The jar will be fully filled, when
three cones full of water is poured
into it. It shows that volume of the
cone is 1/3 x π (r x r) h.

6. A sector of a circle of radius 15cm has
the angle
It is rolled up so that two bonded radii
are joined together to form a cone.
15cm
15cm
Here the radii are joined together. Clearly the radii
of circle is converted to the slant height of cone and
the arc AB is converted to circumference of the cone.

7. To determine the volume of the cone so formed. We first
determine the length of arc AB.
Length of arc AB = 120 / 360 x 2π x 15
= 10π
Suppose the radius of cone be r,
So, 2πr = 10π
or, r = 5cm
Slant height of the cone is 15cm.
Height of the cone = (15) - (5)
= (15+5) (15-5)
= 20 x 10
= 10 2 cm
Volume of the cone = r h
22
1
3
π 2
1
3
2
7
10 2
22
= x x x =5 370.33 cm
3

8. SOLVED EXAMPLES
EXAMPLE 31 : Find the volume of the largest right circular cone that
can be cut out of a cube whose edge is 7 cm.
SOLUTION : For largest right circular cone to be cut, clearly the circle
will be inscribed in a face of the cube and its height will
be equal to an edge of the cube.
Radius of the base of cone, r = cm
So. Volume of the cone = π r h
= x x
x cm
=
7
2
1
3
2
3
1 22
7
7
2
2
7
3
88.83 cm
3

9. SOLVED EXAMPLES
EXAMPLE 3 : The circumference of the base of a 16 m high solid cone is 33 m.
Find the volume of the cone.
SOLUTION : Radius of the base = r m, Height of the cone, h = 16 m
Circumference of the base = 2 π r = 33 .
r =
33
2π
=
π
33
2 X 22 /7
=
21
4
m
Volume of the cone =
1
3
r 2
h
=
1
3
x 22
7
x 21
4
2
x 16
= 22 x 21 = 462 m3