## What is a cone?

A line, whose one end is fixed and the other end is a closed curve is a place, generates a cone. The fixed point is called the vertex or apex.

To create a cone

- Take a circle and a point, called the vertex, which lies above or below the circle.
- Join the vertex to each point on the circle to form a solid.

A **Right cone **is a cone whose vertex lies directly above or below the centre of the circular base. It is a cone whose base is a circle and whose axis is perpendicular to the base. A right circular cone can be constructed by rotating a right-angled triangle 360 degrees about one of the sides other than the hypotenuse.

A perpendicular line dropped from the vertex of the cone to the circular base, is called the height h of the cone.

The length of any of the straight lines joining the vertex to the circle is called the **slant height** of the cone.

s is the slant height which can be found out with the help of Pythagoras formula.

s^{2 }= h^{2} + r^{2 } where r is the radius of the circular base and h is the height of length of the perpendicular line.

## Surface Area of a Cone

Suppose the cone has radius r, and slant height l, then the circumference of the base of the cone is 2πr.

**Altitude: **As defined earlier as the height “h”**, **It is the perpendicular distance from the apex to the circular base of the cone.

### Lateral surface of a Cone

It is the curved surface area. If a hollow cone is made a cut along a straight line from the vertex to the circumference of the base, the cone is opened out and a sector of a circle with radius is produced. Since, the circumference of the base of the cone is 2π r , therefore the arc length of

the sector of the circle is 2π r

Thus,

Lateral surface area of a cone = 1/2 x radius x arc length

= 1/2 x s x 2π r

= πrs

Where

s= slant height ; s^{2 }= h^{2} + r^{2}

Lateral surface = 1/2 of Perimeter of the base x slant height

= 1/2 x 2 x π x r x s

### Total surface area

= Lateral surface area + area of the base

=

2

π r + π r

=

π r( + r)

**Volume of a Right Circular Cone**

If r is the radius of the base, h is the height and is the slant height

Volume of the cone = 1/3 x area of the base x height

= 1/3 π r^{2} h

## Solved Examples

**Find the volume and the total surface area of a cone of radius 6.6cm and height of 12.5cm.**Here r = 6.6cm, h = 12.5cm

**Volume = 1/3 π r**^{2}h

= 1/3 π(6.6)^{2}x 12.5

= 570.199 cu. cm

Since, s^{2 }= h^{2}+ r^{2}

= 12.52^{2}+ 6.62^{2}

= 156.25 + 43.56

s^{2}= 199.81

s = 14.14cm

**Total surface area = π r (s+r)**

=π 6.6 (14.14 + 6.60)

=π 6.6(20.74)

= 430.03 sq. cm.**The height of a cone is 16 cm and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone (Use π = 3.14).**

Solution : Here, h = 16 cm and r = 12 cm.

So, from s^{2 }= h^{2}+ r^{2, }we have

s^{2}= 16^{2 }+ 12^{2}

= 20cms

So, curved surface area = πrs

= 3.14 × 12 × 20 cm^{2}

= 753.6 cm^{2}

Further, total surface area = πrs + πr^{2}

= (753.6 + 3.14 × 12 × 12) cm^{2}

= (753.6 + 452.16) cm^{2}

= 1205.76 cm^{2}**A corn cob (see Fig. 13.17), shaped somewhat****like a cone, has the radius of its broadest end as 2.1 cm and****length (height) as 20 cm. If each 1 cm**of^{2}**the surface of the****cob carries an average of four grains, find how many grains****you would find on the entire cob.**

Since the grains of corn are found only on the curved surface of the corn cob, we would need to know the curved surface area of the corn cob to find the total number of grains on it. In this question, we are given the height of the cone, so we need to find its slant height.Here, s^{2}=r^{2}+h^{2}

= 2.1^{2 }+ 20^{2}

= 404.41 cm

= 20.11 cm

Therefore, the curved surface area of the corn cob = πrs

= 22/7 × 2.1 × 20.11 cm^{2}

= 132.726 cm^{2}

= 132.73 cm^{2}(approx.)Number of grains of corn on 1 cm2 of the surface of the corn cob = 4Therefore, number of grains on the entire curved surface of the cob

= 132.73 × 4 = 530.92 = 531 (approx.)So, there would be approximately 531 grains of corn on the cob.

You can check out solutions of NCERT class 9 Maths here!