Cones Around us:

An ice cream cone

The barriers seen off the road

A conical flask in the chemistry laboratory

A funnel

Birthday caps

As you will be able to relate, all these objects taper from the flat base to the pointed end, called the apex or vertex. A cone is actually formed by a set of line segments or lines connecting the vertex to all the points on the circular base that does not contain the apex. The distance from the base of the cone to its vertex is known as its height. The circular base has a measured radius. The length of the cone from the vertex to any point on the circumference of the base circle is known as the slant height.

Using these fundamentals, we shall arrive at the formulae for the curved and total surface areas of a cone.

A right circular cone is a circular cone whose axis is perpendicular to the circular base and touches its midpoint. Therefore, the center of the circular base when joined with the vertex should form a right-angled triangle with the slant height as the hypotenuse.

In other words, if you rotate a right triangle along any of its legs, the three-dimensional figure so obtained will be called a right circular cone.

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Observe the figure given above.

By Pythagoras theorem,

L2 = h2 + r2

L = \[\sqrt{h^{2} + r^{2}}\]

Where L is the slant height, h is the height of the cone, and r is the radius of the cone.

Let us suppose we have to paint a cone (a birthday cap). While doing so, we can only paint the side area of the cap, not the circular base.

This is what we mean by the curved surface area.

The curved surface area of a cone is the measurement of its outer area, where the base area will not be included.

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Fig(a) is a sector of a circle from which a cone has been made(fig b).

Let the radius of the circle and the length of arc be 'L' and 'c' respectively.

On folding, the radius of the circle ‘L’ becomes the slant height of the cone.

Area of a sector of the circle(curved surface area of the cone)

= \[\frac{\text{arc length of the sector}}{\text{circumference of circle}}\times {\text{area of circle}}\]

=(C/2πL) × πL2

=(2πr/2πL) × πL2

= πrL

Where r is the radius of the base, and L is the slant height of the cone.

In order to obtain the total surface area of any figure, we have to add the curved (lateral) surface area to the area of its top and bottom. In this case, the cone has only one circular surface at the base, which has to be added to the curved surface area to get its total surface area.

Total surface area of cone formula = Curved surface area + area of the base.

As we know, a cone has a circular base. So, the base area of the cone is πr2, hence the formula becomes,

TSA = πrl + πr2

TSA = πr (l+r)

Question 1:- A cone has a radius of 3cm and height of 4cm, find the total surface area of the cone.

Solution:-

First, we need to find the slant height of the cone, which will be determined by using Pythagoras theorem, since the cross-section is a right triangle.

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l2 = h2 + r2

l2 = 42 + 32

l2 = 16 + 9

l2 = 25

l = 5cm

Total surface area of cone (TSA of cone) = πr(l+r)

= π.3 (5+3)

= π.3.8

= 24π cm2

Therefore, the total surface area of the cone is 24π cm2.

Question 2:- The slant height of a cone is 13cm. the diameter of the base is 10cm. Find the total surface area of the cone.

Solution:- Given that,

Slant height: l = 13cm

Diameter: d = 10cm

Radius: r = d/2 = 10/2 = 5cm

Curved surface area = πrl

= π.5. 13

= 65π cm2

Total surface area = πr (l+r)

= π.5(13+5)

= π.5.18

= 90πcm2

Therefore the total surface area of the cone is 90πcm2.

Volume of a cone = \[\frac{1}{3}\]πr2h

The volume of a frustum = \[\frac{1}{3}\] πh (r2+Rr+R 2) where r and R are the two different radii of the respective bases.

Curved surface area of a frustum = πl(R+r) where l is the slant height of the frustum

Total surface area of a frustum = πl(R+r) + πr2+ πR2

FAQ (Frequently Asked Questions)

1) How Many Types of Cones are There?

Solution: There are two types of cones- right circular cone and oblique cone.

In a right circular cone, the base and the axis form a right angle. This is the most commonly used cone in geometry.

An oblique cone has a circular base, but the vertex of the cone is not directly situated above the center of the circular base.

2) What is the Frustum of a Cone?

Solution: Frustum of a cone is obtained when a right circular cone is cut in a manner such that the base of the original cone and the plane along which the cut is made is parallel to each other. Therefore, the frustum of a cone has two circular bases on the two ends, though both are of different radii.

The commonly used glass at home for drinking water is a good example of a frustum. To calculate the total surface area of the frustum, we will have to add the areas of both these circular surfaces to the curved surface area of the frustum. The figure given below shows a frustum.

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