A cone is a three-dimensional solid having a circular base and the apex or vertex. The Right Circular Cone is the cone in which the line of the axis is perpendicular to the base. It has a flat surface and a curved surface. The curved surfaces meet at the point known as the vertex or apex of the cone.

In our daily life, we can see various objects that have the shape of a right circular cone, like ice-cream cones, birthday celebration caps, waffle cones, etc. In this article, we will learn all about the right circular cones.

Definition of Right Circular Cone

The right circular cone is the three-dimensional solid. A right circular cone is a cone in which the line of the axis is perpendicular to the base. In other words, a cone in which a line joining the vertex to the center of the cone is perpendicular to the base of the cone is called a right circular cone.

A right circular cone is formed by revolving the right-angled triangle about any one of its sides other than the hypotenuse. The base of the cone is in a circular shape. A right circular cone has a flat surface and curved surface, and they meet up at a point known as the vertex or apex of the cone.

The right circular cone with radius \((r)\) of the circular base, height of the cone is \(h\), and slant height of the cone is \(l\) as shown below:

Terms of a Right Circular Cone

There are few terms associated with a right circular cone. There are mainly three terms of the cone. These are radius, height, and slant height. Observe the below given right circular cone:

Vertex:

It is the point of contact of all the curved surfaces area of the cone. The fixed point of the cone, where the curved lines are joined, is known as the vertex of the cone. It is also known as the apex of the cone.
In the below figure, \(A\) is the vertex of the right circular cone.

Radius:

The distance of any point on the circumference of the base (circle) to the center of the base is known as the radius of the base of the right circular cone.

Height:

The distance between the apex or vertex to the center of a base is known as height of the cone.
In the above figure, the line segment \(OA\) is known as the height of the right circular cone.

Slant Height:

The distance between the vertex to any point on the circumference of the base of the cone is called slant height.
In the above figure, the line segment \(AB\) is known as the slant height of the right circular cone.

Properties of a Right Circular Cone

The certain properties of the right circular cone are discussed below:

1. The base of the right circular cone is in a circular shape.
2. A right circular cone has one vertex.
3. Axis or line joining a vertex to the centre of the base is perpendicular to the base.
4. The slant height is the distance from the vertex to the edge of the base of the right circular cone. It is denoted by “\(l\)” or “\(S\)”
5. The altitude of the right circular cone coincides with an axis. It is denoted by \(h.\)
6. A right circular cone is formed by rotating the right-angled triangle about any one of the sides of the triangle other than the hypotenuse.
7. The surface area thus formed by the hypotenuse gives the curved surface area or lateral surface area of the right circular cone.
8. A horizontal section parallel to the base of the right circular cone gives the cross-section of the cone.

Slant Height of Right Circular Cone

The distance between the vertex to any point on the circumference of the base of the cone is called slant height. The line segment \(AB\) is the slant height of the right circular cone.
Consider a right circular cone of radius \(r\), height \(h\), and slant height \(l\);

From the above figure, \(∆ AOB\) is the right triangle.
By Pythagoras theorem: \(\text{hypotenuse}^2 = \text{side}^2 + \text{side}^2\)
In \(∆ AOB\),
\( \Rightarrow {\text{A}}{{\text{B}}^2} = {\text{O}}{{\text{A}}^2} + {\text{O}}{{\text{B}}^2}\)
\( \Rightarrow {l^2} = {r^2} + {h^2}\)
\(\therefore \,l = \sqrt {{r^2} + {h^2}} \)
By using the above formula, we can find the slant height of the right circular cone with the help of height and radius.

Surface Area of Right Circular Cone

The surface area of the right circular cone is the region of space occupied by the surface of the right circular cone. The surface area of the right circular cone is expressed in the square units like \(\rm{m}^2\), \(\rm{cm}^2\), \(\rm{in}^2\) and \(\rm{ft}^2\) etc.

To observe the surface area of the right circular cone, cut the right circular cone and open it as shown in the below figure:

If we cut a right circular cone, we will get a sector with a radius equals to slant height \((l)\). There are two types of the surface area of the right circular cone are
1. Curved surface area
2. Total surface area

Curved Surface Area

Space or region occupied by a curved surface of the right circular cone is called the curved surface area or lateral surface area of the right circular cone. It is written in short as C.S.A. The curved surface area of the right circular cone does not include the area of the base.

The curved surface area of the right circular cone equals the area of the sector formed (as explained above).
\( {\text{C}} {\text{.S}} {\text{.A}} = \pi rl\)

Total Surface Area

Space or region occupied by a complete surface of the right circular cone is called the total surface area. In short, it can be written as T.S.A. The total surface area of the right circular cone is the sum of the areas of the curved surface area of the cone and the area of the base.

T.S.A \(=\) Area of base \(+\) curved surface area of the cone.
\( {\text{T}} {\text{. S}} {\text{.A}} = \pi  {r^2} + \pi rl\)

Here,
\(r-\) radius of the right circular cone
\(l-\) slant height of the right circular cone

Note: Total surface area of a right circular cone is referred to as the surface area of a right circular cone. Thus, in problems, to calculate the surface area means we need to find the total surface area of the right circular cone.

Volume of a Right Circular Cone

The volume of a right circular cone is the space occupied by a cone in the given plane. The volume of a right circular cone is expressed in cubic units like \(\rm{cm}^3\), \(\rm{m}^3\) etc. It can be expressed using the units litres.

The volume of the right circular cone equals one-third of the product of the area of base and height of the right circular cone.
\(\text{Volume} = \frac{1}{3} \times \text{area of base} (\text{circle}) \times \text{height}\)
\(\text{Volume} = \frac{1}{3} \times \pi {r^2}h\)
The volume of a right circular cone, when its base diameter is given as follows:

We know that diameter equals twice the radius \((d = 2r)\)
\( r = \frac{d}{2}\)
\( \Rightarrow {\text{V}} = \frac{1}{3} \times \pi \times {\left({\frac{d}{2}} \right)^2} \times h\)
\( \Rightarrow {\text{V}} = \frac{1}{{12}}\pi {d^2}h\)
Here,
\(r-\) radius of the base
\(d-\) diameter of the base
\(h-\) height of the right circular cone
The below figure shows how the volume of the right circular cone equals one-third of the volume of the cylinder.
The volume of the right circular cone \(= \frac{1}{3} \times {\text{volume of a right cylinder}}\)

Right Circular Cone Equation

The equation of the right circular cone whose vertex is at the origin is given below
\(\left({{x^2} + {y^2} + {z^2}}  \right){\cos ^2}\theta  = {(lx + my + nz)^2}\)
Here,
\(θ-\) Semi vertical angle
\((l, m, n)\) direction cosines of the axis of the right circular cone.

Solved Examples – Right Circular Cone

Q.1. Find the curved surface area of the right circular cone whose radius is \(2\) units and height is \(2\sqrt 3\) units.
Ans: Given, the radius of the right circular cone \((r) = 2\) units.
Height of the right circular cone \((h) = 2\sqrt 3 \) units.

The slant height of the right circular cone is found by using the formula: \(l = \sqrt {{r^2} + {h^2}} \)
\( \Rightarrow l = \sqrt {{2^2} + {{(2\sqrt 3 )}^2}} \)
\( \Rightarrow l = \sqrt {4 + 12} \)
\( \Rightarrow l = \sqrt {16} = 4\) units

We know that the curved surface area of the right circular cone is \(πrl\).
\( \Rightarrow {\text{C}}.{\text{S}}.{\text{A}} = \frac{{22}}{7} \times 2 \times 4\)
\( \Rightarrow {\text{C}}.{\text{S}}.{\text{A}} = \frac{{176}}{7}{\text{sq}}{\text{.units}}\)
Hence, the curved surface area of the right circular cone is \(\frac{{176}}{7}{\text{sq}}{\text{.units}}\)

Q.2. Find the volume of the tent, which is in the form of a right circular cone, whose radius and height are \(24\,\rm{m}\) and \(10\,\rm{m}\) respectively.
Ans: Given that, a tent is in the form of a right circular cone.
And, the radius of the tent is \(r = 24\,\rm{m}\), height of the tent \(h = 10\,\rm{m}\)

We know that volume of the right circular cone is given by \(\frac{1}{3}\pi {r^2}h\)
\({\text{V}} = \frac{1}{3} \times \frac{{22}}{7} \times {24^2} \times 10\)
\( \Rightarrow {\text{V}} = \frac{{42,240}}{7}{{\text{m}}^3}\)
Hence, the volume of the tent is \(\frac{{42,240}}{7}{{\text{m}}^3}\)

Q.3. A child is playing near the bank of the river. He made a sand cone with a radius \(5\,\rm{cm}\) and a height \(12\,\rm{cm}\). Find the amount of sand used in the cone.[Take \(π = 3.14\)]
Ans: The radius of the cone \((r) = 5\,\rm{cm}\).
Height of the cone \((h) = 12\,\rm{cm}\)

We know that the volume of the right circular cone is \(\frac{1}{3}\pi {r^2}h\)
\({\text{V}} = \frac{1}{3} \times 3.14 \times {5^2} \times 12\)
\({\text{V}} = 314\,{\text{c}}{{\text{m}}^3}\)
Hence, the amount of sand inside the cone is \( 314\,{\text{c}}{{\text{m}}^3}\)

Q.4. Latha prepares a birthday cap with a piece of paper in the form of a right circular cone radius \(3\) inches and height \(4\) inches. Find the slant height of the birthday cap made by her.
Ans: Given that, the birthday cap prepared by Latha is in the shape of a right circular cone.

The radius of the cone \((r) = 3\) inches and height of the cone \((h) = 4\) inches.
We know that the slant height of the cone is given by \(l = \sqrt {{r^2} + {h^2}} \)
\( \Rightarrow l = \sqrt {{3^2} + {4^2}} \)
\( \Rightarrow l = \sqrt {{9} + {16}} \)
\( \Rightarrow l = \sqrt {25} = 5\) inches
Hence, the slant height of the birthday cap is \(5\) inches.

Q.5. Find the total surface area of the cone with base radius \(7\,\rm{m}\) and slant height \(15\,\rm{m}\).
Ans: Given base radius of the cone is \((r) = 7\,\rm{m}\).
The slant height of the right circular cone is \((l) = 15\,\rm{m}\).
We know that the total surface area of the right circular cone is \(πrl(r + l)\)
\( \Rightarrow {\text{T}}.{\text{S}}.{\text{A}} = \frac{{22}}{7} \times 7 \times 15 \times (7 + 15)\)
\( \Rightarrow {\text{T}}.{\text{S}}.{\text{A}} = 22 \times 15 \times 22\)
\( \Rightarrow {\text{T}}.{\text{S}} \cdot {\text{A}} = 7260\,{{\text{m}}^2}\)
Hence, the total surface area of the given cone is \(7260\,{{\text{m}}^2}\)

Summary

In this article, we have studied the definition of cone and the right circular cone, and we discussed real-life examples of the right circular cone. We also studied properties of the right circular cone and terms related to the right circular cone-like vertex or apex, axis, base, radius, height and slant height etc.

Here, we studied the definitions of the area, the volume of the right circular cone, and its formulas. We discussed the different types of formulas to calculate slant height, curved surface area or lateral surface area, total surface area, and volume of the right circular cone with solved examples.

Frequently Asked Questions (FAQ) – Right Circular Cone

Q.1. What is the formula for the total surface area of the right circular cone?
Ans: The formula used to calculate the total surface area of the right circular cone with radius \((r)\) and slant height \((l)\) is \(πrl(r + l)\).

Q.2. How many surfaces are there in a right circular cone?
Ans: A right circular cone has two surfaces.

Q.3. How do you find the length of the right circular cone?
Ans: We can find the length (slant height), the height of the right circular cone by using the formula
\(l = \sqrt {{r^2} + {h^2}} \)

Q.4. What is the formula for the volume of the right circular cone?
Ans: The formula used to calculate volume \(\frac{1}{3}\pi {r^2}h\)

Q.5. How many vertices are there in a right circular cone?
Ans: A right circular cone has only one vertex.

We hope this detailed article on the right circular cone topic has helped you in your studies. If you have any doubts or queries regarding this topic, you can comment down below and we will be ready to help you.