CONEQ. 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.

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.

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 + r

2 2

DERIVATIONVOLUME OF A RIGHT

CIRCULAR CONEVolume of a cone = 1/3 x π (r x r) hThe above formula can be verifiedexperimentally. Take a cylindricaljar of height h and radius r whosevolume 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.

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.

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

13

π 2

13

2

7 10 2 22= x x x =

5 370.33

cm3

SOLVED EXAMPLESEXAMPLE 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

=

7213

2

31 22

772

27 3

88.83 cm3

SOLVED EXAMPLESEXAMPLE 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

332 X 22 /7 =

214

m

Volume of the cone

=13

r 2 h

=13 x 22

7x 21

4

2x 16

= 22 x 21 = 462 m3