Solid Cone

In mathematics, we come across various shapes and figures and try to apply them in real life. This article focuses on one such shape, a solid cone. Today we will cover the properties and applications which come under a solid cone.

A cone is a geometric shape, three dimensional in nature, which gradually from a circular base, tapers smoothly to a single point called the vertex or apex. Frequently, although not necessarily, they form circular shapes.

In elementary geometry, we often assume cones to be the right cones. Right cones are conical shapes that have their vertex directly over the centre of the base. We know other cones with unique positions of the vertex as oblique cones.

Types of Cones? 

Categorically, there are two types of cones. 

• Right Circular Cone
• Oblique Cone

The position of the vertex defines these cones. A vertex is a common point, all the line segments of a cone connect with.

Right Circular Cones are those cones that have their vertex right opposite the base of the cone. The line that passes through the base of the cone is perpendicular to the radius is the height of the cone.

Oblique Cones are those cones that do not have their vertex directly opposite the centre of the base of the cone. In this case, the line that passes through the centre of the base as the height is not perpendicular to the radius. Cones may shape up to be right or oblique with respect to the position of the vertex. However, cones will always follow these three principles, which are covered under their properties:

1. Cones have circular bases

Cones have a circular base. This property comes in use if we are asked about the area or the volume of the base of a cone. The base of the cone follows the same formulae a circle has.

2.It has only one Edge (Vertex): A solid shape of a cone may appear to have 1 vertex in the form of a corner, but the cone has no corners. A solid cone has 1
edge, 1 face and zero corners.

3.A Cone has only 1 face. We believe ‌a cone has 2 faces, one as the curved surface and one as the circular base. However, a cone has only one curved face that is taken into consideration.

Centre of Mass of a Solid Cone

We can define the centre of mass of an object as a point in which the whole of the mass of the body appears to be concentrated towards. With a cone, the point at which it equally distributes the entire weight of the cone is called the point of the centre of mass.

The centre of mass is located along a line that is perpendicular to the base of the cone. It is often 1/4th the height of the cone. 

Moment of Inertia of a Cone

Moment of inertia is the property of a body to resist angular acceleration. It is the sum of the particles present in the mass of an object with the square of the distance from the axis of rotation.

We can calculate moment of inertia for a right circular cone by taking an integral over the volume of the cone and weighting each unit of its mass by its distance from the axis squared.

Formulae of Solid Cones

Solid cones have varied applications in ‌mathematics. These applications have only come this far with the help of the formulae derived from the shape of the cone.

V= Volume
R= Radius
h= Height

l= slant height

Surface Area of a Cone

The surface area of a cone is r (r+s)

Volume of a Cone

The formula for volume of a cone is: 1/3πr²h

Height of a Cone

The height of a cone is h= 3V/R²

Slant height of a cone?

The formula for calculating the slant height of a cone is 

l² = r² + h²

Conclusion

A cone is a figure that has a circular base as well as a curved surface area. 

• A cone has a circular base.
• There are no edges in a cone.
• The length of the line segment joining the apex of the cone to the base of a cone is the slant height of that respective cone.

Categorically, there are two types of cones. Right Circular Cone and an Oblique Cone. These cones are defined by the position of the vertex. A vertex is a common point for all the line segments of a cone connected with.