Cone: A cone is a three-dimensional solid geometrical object having a circular base and a pointed edge at the top called the apex or vertex. It has one curved surface and one circular base, one vertex, and one edge.

What is the Shape of Cone?

A cone is a shape formed by using a set of line segments that connect a common point, called the vertex, to all the points of a circular base. The distance from the vertex of the cone to the base is the height \(h\) of the cone. The circular base has a radius \(r\). The length of the cone from apex to any point on the circumference of the base is the slant height \(l\).

What are the Cone Formulae?

To derive the formula of surface area and volume of a cone, one should know the slant height. The formula for the slant height is derived using the Pythagoras theorem.

In the above figure, consider a right-angled triangle \(ABC\), in which \(AB = h,BC = r\) and \(AC = l\). Applying Pythagoras theorem to triangle \(ABC\), we get:

\({(AC)^2} = {(AB)^2} + {(BC)^2}\)

\( \Rightarrow {l^2} = {h^2} + {r^2}\)

\( \Rightarrow l = \sqrt {\left( {{h^2} + {r^2}} \right)} \)

All the important formulas related to cone are tabulated below:

➯ Volume \(\left( V \right)\) of a cone with a radius \(\left( r \right)\) height \(\left( h \right)\) is given by: \(V = \frac{1}{3}\pi {r^2}h\) ➯ Lateral surface area of a cone is given by \(\pi rl.\) ➯ Total surface area of the cone is the sum of lateral surface area and the area of the circular base. Therefore, the total surface area of cone \( = \pi rl + \pi {r^2}\) \( = \pi r(l + r)\)

What are the Cone Properties?

1. A cone has one curved surface and a circular base.
2. Cone has one edge.
3. Cone has only one vertex or apex point.
4. The slant height of a cone is \(l = \sqrt {\left( {{h^2} + {r^2}} \right)} \)
5. The volume of a cone is given by \(V = \frac{1}{3}\pi {r^2}h\)
6. The total surface area of a cone is given by \(\pi r(l + r)\)

What are the Types of a Cone?

There are two types of cones.

1. Oblique Cone
2. Right Circular Cone

Oblique Cone

A cone whose surface is caused by line segments joining a fixed point to the points of a circle, and the fixed point lying on a line is not perpendicular to the circle at its centre is known as an oblique cone.

That is, if the axis of the cone is not perpendicular to the plane of the circular base, then the cone is an oblique cone. 

Right Circular Cone

A right circular cone is a cone with a circle as its base and a line segment perpendicular to the base passing through its centre and passing through the vertex or apex of the cone formed by turning a right-angled triangle along one of the sides is called a right angle cone.

That is, if the axis of the cone is perpendicular to the plane of the circular base, then the cone is a right circular cone. 

Solved Examples – Cone

Question 1: Find the volume of a cone if radius \(r = 6\;{\rm{cm}}\) and height \(h = 7\;{\rm{cm}}\).
Consider \(\pi  = \frac{{22}}{7}\).

Answer: Given: \(r = 6\;{\rm{cm}}\) and \(h = 7\;{\rm{cm}}\)
We know that, the volume of a cone \(V = \frac{1}{3}\pi {r^2}h\)
\( \Rightarrow V = \left( {\frac{1}{3} \times \frac{{22}}{7} \times {6^2} \times 7} \right){\rm{c}}{{\rm{m}}^3}\)
\( \Rightarrow V = \left( {\frac{1}{3} \times 22 \times 36} \right){\rm{c}}{{\rm{m}}^3}\)
\( \Rightarrow V = 264\;{\rm{c}}{{\rm{m}}^3}\)
Therefore, the volume of a cone is \(264\;{\rm{c}}{{\rm{m}}^3}\)

Question 2: What is the total surface area of the cone whose radius is \(r = 4\,{\text{cm}}\) and height is \(h = 6\,{\text{cm}}\) ?
Consider \(\pi  = \frac{{22}}{7}.\)

Answer: Given \(r = 4\,{\text{cm}}\) and \(h = 6\,{\text{cm}}\)
We know that the total surface area of a cone is \(\pi r(l + r)\)
Also, \(l = \sqrt {\left( {{h^2} + {r^2}} \right)} \)
\( \Rightarrow l = \sqrt {\left( {{6^2} + {4^2}} \right)} \)
\( \Rightarrow l = \sqrt {52} \)
\( \Rightarrow l = 7.211\;{\rm{cm}}\)
Now, the total surface area of cone \(\pi r(l + r)\)
\( = \frac{{22}}{7} \times 6 \times (7.211 + 4)\)
\( = 211.215\)
Therefore, the total surface area of the cone is \(211.215\;{\rm{c}}{{\rm{m}}^2}.\).

Question 3: Find the curved surface area of a right circular cone whose slant height is \(10\,{\text{cm}}\) and base radius is \(7\,{\text{cm}}\)
Consider \(\pi  = \frac{{22}}{7}\)

Answer: We know that the curved surface area of a cone \( = \pi rl\)
Given: \(l = 10\,{\text{cm}}\) and \(r = 7\,{\text{cm}}\)
So, the curved surface area of a cone \( = \frac{{22}}{7} \times 7 \times 10\,{\text{c}}{{\text{m}}^2}\)
\( = 220\,{\text{c}}{{\text{m}}^2}\)
Therefore, the curved surface area of a cone \( = 220\,{\text{c}}{{\text{m}}^2}\)

Question 4: The height of a cone is \(16\,{\text{cm}}\) and its base radius is \(12\,{\text{cm}}\). Find the curved surface area and the total surface area of the cone (Use \(\pi  = 3.14\)).

Answer: Given: \(h = 16\,{\text{cm}}\) and \(r = 12\,{\text{cm}}\) We know that the curved surface area of a cone\( = \pi rl\) and the total surface area of a cone \( = \pi r\left( {l + r} \right)\)
First, we need to find the slant height \(l.\)
We know that, slant height \(l = \sqrt {\left( {{h^2} + {r^2}} \right)} \)
\( \Rightarrow l = \sqrt {\left( {{{16}^2} + {{12}^2}} \right)} \)
\( \Rightarrow l = \sqrt {256 + 144} \)
\( \Rightarrow l = \sqrt {\left( {400} \right)} \)
\( \Rightarrow l = 20\,{\text{cm}}\) Now, the curved surface area of cone \( = 3.14 \times 12 \times 20\,{\text{c}}{{\text{m}}^2} = 753.6\,{\text{c}}{{\text{m}}^2}\)
The total surface area of cone \( = 3.14 \times 12\left({20 + 12} \right)\,{\text{c}}{{\text{m}}^2}\)
\( = 3.68 \times 32\,{\text{c}}{{\text{m}}^2}\)
\( = 1205.76\,{\text{c}}{{\text{m}}^2}\)
So, the total surface area of the cone \( = 1205.76\,{\text{c}}{{\text{m}}^2}\)

Question 5: A bus stop is barricaded from the remaining part of the road, by using \(50\) hollow cones made of recycled cardboard. Each cone has a base diameter of \(40\,{\text{cm}}\) and a height \(40\,{\text{cm}}\). If the outer side of each of the cones is to be painted and the cost of painting is \(12\) per \({{\text{m}}^2}\) what will be the cost of painting all these cones? (Use \(\pi  = 3.14\) and take \(\sqrt {1.04}  = 1.02\)

Answer: Given: \(h = 1\,{\text{m,}}\) diameter \(d = 40\,{\text{cm}}{\text{.}}\)
So \(r = \frac{d}{2}\, = \frac{{40}}{2}{\text{cm=0}}{\text{.2}}\,{\text{m}}\)
From the question, we can say that the curved surface area of a cone will be painted.
We know that the curved surface area of a cone \( = \pi rl\)
Also, slant height \(l = \sqrt {\left( {{h^2} + {r^2}} \right)} \)
\( \Rightarrow l = \sqrt {\left( {{1^2} + {{\left( {0.2} \right)}^2}} \right)} \)
\( \Rightarrow l = \sqrt {1.04} \)
\( \Rightarrow l = 1.02\,{\text{m}}\)
The curved surface area of one cone \( = 3.14 \times 0.2 \times 1.02\,{{\text{m}}^2}\)
\({\text{=0}}{\text{.64046}}\,{{\text{m}}^2}\)
So, the curved surface area of \( = 50 \times 0.64046\,{{\text{m}}^2}\)
\( = 32.028\,{{\text{m}}^2}\)
Cost of painting \(1\,{{\text{m}}^2} = 12\)
So, the cost of painting \(32.028\,{{\text{m}}^2} = 12 \times 32.028\)
\( = 384.336\)
Therefore, the cost of painting \(50\) cones is \( = 384.34.\)

Summary

From this article, we have learned what is the shape of a cone, the definition of a cone, elements of a cone, the properties and different types of cone, the formulas to find the curved surface area, slant height, total surface area, and volume of a cone. Also, we solved some example problems on cones.

Frequently Asked Questions (FAQ) – Cone

Question 1: What is a cone and write its formula?
Answer: A cone is a typical three-dimensional geometric figure that has a flat surface and a curved surface pointed towards the top.

1. Volume of a cone \(V = \frac{1}{3}\pi {r^2}h\)
2. Curved surface area (CSA) of a cone \( = \pi rl\) where \(l = \sqrt {\left( {{h^2} + {r^2}} \right)} \)
3. Total surface area of cone \( = \pi r(l + r)\)

Question 2: What is the curved surface of the cone also called?
Answer: The curved side of cone is also called as lateral surface.

Question 3: What is the shape of a cone?
Answer: A cone is a shape formed by using a set of line segments that connect a common point, called the vertex, to all the points of a circular base.

Question 4: What is the difference between vertices and edges?
Answer: A vertex is a corner where the edges meet. A point where two faces meet is known as an edge.

Question 5: Does a cone have \(1\) or \(2\) faces?
Answer: A cone has \(2\) faces. It has one curved face and a circular base.

Question 6: How many edges are there in a cone?
Answer: The cone has only one edge formed by the joining of a circular base and curved surface.

Question 7: Can a cone roll?
Answer: A cone will roll on its curved surface in a circular path with vertex as the centre and slant height as the radius. 

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