Right Circular Cone

‘Cone’ comes from the Greek word ‘konos,’ which means a peak or wedge. We come across several three-dimensional shapes like ice cream cones, a clown’s cap, a tent, and so on in our daily lives. Did you ever wonder how much ice cream it takes to fill a cone of ice cream? Isn’t it true that the cone resembles a triangle with a bent surface? So, how does a conical shape come about? What is a right circular cone, and how to calculate its volume and surface area? Read on to discover solutions to these questions.

Definition of a Cone

A cone is a three-dimensional geometric object with a flat bottom surface and a curving top surface that points upward. The cone’s pointy end is known as the apex or vertex. The flat surface is circular and known as the base. Cones can be seen in a wide range of daily items. The following are some examples:

• A funnel is conical in shape.

• Ice-cream cones

• Conical barriers on the roads.

• Regular birthday and clown hats are conical in shape.

Characteristics

The following are the three primary characteristics of a cone:

• A cone has only one round surface at the bottom and a curved lateral surface on the sides.

• There are no edges in a cone.

• It has one corner known as the vertex.

Cones come in a variety of shapes and sizes. Depending on the position of the vertex on the base, a cone can fall into one of two categories:

1. Right circular cone – A right circular cone has a perpendicular vertex to the base, meaning that the perpendicular line passes through the centre of the cone’s circular base. For example, a birthday hat is a typical right circular cone.

2. Oblique cone – It is a type in which the vertex is located somewhere other than the base’s centre. Oblique cones are mostly utilised in industry for conveying and storing commodities, ventilation systems, and huge pipeline transitions.

Now let’s focus on understanding what is a right circular cone, which is also the primary topic of this article.

Right Circular Cone

Right circular cones are three-dimensional objects whose axis is perpendicular to the circular base. In other words, the perpendicular axis makes an angle of 900 with the radius of the base. The base of this cone is shaped like a circle. A revolving right triangle about one of its perpendicular axes produces a right circular cone. The radius, perpendicular height, and slant height are the three segments that make up a right circular cone. Following are the basic properties of the right circular cone:

• The radius is the horizontal distance from the centre of the circular base to any point on its circumference. The radius is usually denoted by the small letter ‘r’.

• The height is the distance between the apex of the cone and the centre of the circular base. The height is usually denoted by the small letter ‘h’.

• The slant height is the distance between the apex and any point on the circumference of the circular base. The slant height is usually denoted by the small letter ‘l’.

• It has a round base. The axis is a line that runs from the vertex to the base’s centre.

• Rotating a triangle forms a right circular cone with the perpendicular side as the axis of rotation. The triangle’s hypotenuse creates a curved surface.

Basic Formulas

A right circular cone formula describes the relationships between the radius(r), height(h) and slant height(l). Understanding the relationships will help you derive equations for the surface areas and volume of the geometrical object. For easy calculations, consider a right circular cone with the below parameters:

• The height of the cone is ‘h.’

• The radius of the circular base is ‘r.’

• The slant height of the cone is ‘l’.

Slant height

In a right circular cone, the angle of contact between the height and the radius is 900. Hence, according to Pythagoras theorem, the slant height is:

Hypotenuse² = Perpendicular² + Base²

So,

l² = r² + h² 

where

l = slant height or the hypotenuse

r = radius

h = height

Using the above relation, one can calculate any variables using the other two given values.

Surface Area

Surface area is termed the total area covered by the object’s surface. It is measured in square units such as m², cm², etc. The surface area is categorised into two types:

1. Curved Surface Area – Cut open a cone along its perpendicular axes and spread it on a flat surface. The region occupied by the lateral plane is known as the curved surface area(CSA). The CSA excludes the area of the rounded base. The formula for calculating CSA is:

Curved Surface Area = ½ ×  Circumference of the base ×  Slant height

where,

Circumference of the base = 2 × π ×  r

Slant height = l

CSA = ½ ×  2 × π ×  r ×  l

which can also be denoted as CSA = π  ×  r ×  l

Hence, the curved surface area is equal to 3.14 times the product of the radius and slant height of the cone.

1. Total Surface Area – The total surface area(TSA) is the sum of the curved surface area and the area of the circular base. The formula for calculating TSA is:

Total Surface Area = CSA + Area of the circle

where,

CSA = π  ×  r ×  l

Area of the circle =  π ×  r²

TSA = (π ×  r ×  l) ×  ( π ×  r²)

Hence, the total surface area is  π ×  r ×  (r + l).

Volume – The entire space the object occupies in a three-dimensional plane is defined as volume. A cone’s volume is measured in cubic units, such as cm3, m3, etc. The base radius and height of the cone are used to calculate its volume. 

Volume of a cone = (⅓) × π ×  r ×  r ×  h where,

radius of the base = r

height = h

Volume = (⅓) × π × r² × h

Conclusion

Let us now summarise what is a right circular cone? Right circular cones are three-dimensional geometrical objects having a round base, a pointy vertex and a lateral surface. A cone consists of a lateral height, a radius and height to calculate its volume and surface areas. The surface area consists of the curved surface and the total surface area. A few examples of conical solids are party hats, ice-cream cones and funnels. Volcanoes are one of the most well-known cone-shaped objects in nature.