Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is a shape formed by using a set of line segments or the lines which connects a common point, called the apex or vertex, to all the points of a circular base(which does not contain the apex). The distance from the vertex of the cone to the base is the height of the cone. The circular base has measured value of radius. And the length of the cone from apex to any point on the circumference of the base is the slant height. Based on these quantities, there are formulas derived for surface area and volume of the cone. In the figure you will see, the cone which is defined by its height, the radius of its base and slant height.

The three main properties of a cone are: 

  • A cone has only one face, which is the circular base but no edges
  • It has one circular face.
  • It has zero edges.
  • It has one vertex (corner).

The three main elements of a cone are its radius, height, and slant height. 

Radius of the Cone

Radius is defined as the distance between the center of the circular base to any point on the circumference of the base.

Height of the Cone

The height is the distance between the apex of the cone to the center of the circular base. 

Slant Height of the Cone

The slant height of the cone is the distance from the top of the cone to the point on the outer edge of the circular base. The formula for the slant height is derived using the Pythagorean theorem.

Types of Cones 

A cone can be of two categories, depending upon the position of the vertex on the base: 

  • A right circular cone is one whose apex is perpendicular to the base. Here, the axis makes a right angle. 
  • If the vertex position is anywhere besides the center of the base, it is an oblique cone. Here, the axis is non-perpendicular. 
A right circular cone and an oblique circular cone

 

The formula for the surface area and volume of the cone is derived here based on its height(h), radius(r) and slant height(l).

Slant Height : l = √(r2+h2)

Volume of the Cone : V = ⅓ πr2h cubic units

Surface Area of the Cone :

The surface area of a right circular cone is equal to the sum of its lateral surface area(πrl) and surface area of the circular base(πr2). Therefore,

The total surface area of the cone = πrl + πr2

Or

Area = πr(l + r)

Frustum of Right Circular Cone

Frustum of a cone is a piece of the given circular or right circular cone, which is cut in a manner that the base of the solid and the plane cutting the solid are parallel to each other. Based on this, we can calculate the surface area and volume also. 

Properties of Frustum of Cone

Properties of a Frustum of a cone are very similar to the cone, some of the important properties of frustum of cone are,

  • Base of the cone the original cone is contained in the frustum of a cone but its vertex is not contained in the frustum.
  • Formulas of frustum of a cone are dependent on its height and two radii (corresponding to the top and bottom bases).
  • Height of the frustum of the cone is the perpendicular distance between the centers of its two bases.

Formulas of Frustum of Cone

Volume of Frustum of Cone

Frustum of cone is a sliced part of a cone, where a small cone is removed from the larger cone. Therefore, to calculate the volume of the frustum of cone, one just needs to calculate the difference between the volume of the larger and smaller cone. 

 

Volume of the frustum of cone

 

Let’s assume,

  • Total height of the cone is to be H + h
  • Total slant height to be l’ + L
  • The radius of a complete cone is r
  • The radius of the sliced cone is r’

Volume of the frustum of cone = 1/3 πH(r2 + r’2 + rr’)

Surface Area of Frustum of Cone

The surface area of frustum of cone can be calculated by the difference between the surface area of the complete cone and the smaller cone (removed from the complete cone).Surface Area of Frustum of Cone

 

Curved surface area of frustum of cone = πl (r + r’)

Total surface area of the frustum of cone = Curved surface area of the frustum of cone + surface area of the top base + surface area of the bottom base

Total surface area of frustum of cone = πl(r + r’) + π (r2 + r’2

or

Total surface area of frustum of cone = πl (r2 – r’2)/r  + π (r2 + r’2

l is the slant height of the smaller cone that can be given as l = √ [H2 + (r – r’)2]