Cone, Sphere and Cylinder

Any object that occupies space is known as a solid shape or 3-dimensional object. Almost all the objects around us can be considered a 3D shape. In this article, we will learn about three-dimensional figures. We will discuss the area of the cone cylinder and sphere. We will also learn how to derive the curved surface area as well as the total surface area of these solids. 

Volume of Cone and Cylinder

The volume of a cone can be defined as the space or the capacity of the cone. A cone is a three-dimensional geometric shape having a circular base that tapers from a flat base to a point called an apex or vertex. 

Let the radius of the circular base be ‘r’ and its vertical height be ‘h’. Then slant height ‘l’ can be calculated using Pythagoras Theorem.

Using Pythagoras Theorem, we can write

h² + r² = l²

l = \[\sqrt{h^{2} + r^{2}}\] 

The volume of a cone formula is given as ⅓ πr²h cubic units.

Surface Area of Cone Derivation

Area of a Cone

There can be two types of surface areas that we may be concerned about.

• Curved Surface Area - It is defined as the area of the curved portion

• Total Surface Area - It is defined as the area of the curved portion and area of the circular base

Take a paper cone and we have to cut it along its slant height. Observe the figure which is formed by the surface of the cone. The figure obtained will be some portion of a circle. The radius of this circular region will be the slant height ‘l’ of the given cone.

If we divide this figure into infinite parts, (by drawing a straight line from the centre to the circumference) each of the parts will be a triangle 

Now, the total area of this figure is the sum of all of these individual triangles. Hence,

Area of figure = (1/2) × slant height ×  (b₁ + b₂ + b₃ + ………….. + b_n)

                         =  (1/2)× l × (length of an entire curved boundary)

Now, length of entire curved boundary = circumference of the base 

                                                                      = 2πr

Thus, the area of a figure (CSA of a cone) =  (1/2)× l × (2πr)

                                                              = πrl

Total Surface Area (TSA)    = Curved Surface Area(CSA) + Area of circular base

                                               =  πrl + πr²

                                             = πr(l + r)

Cylinder:

A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, that are at a fixed distance.

The formula for the volume of a cylinder is πr²h cubic units.

Where r is the radius of the cylinder 

h is the height of the cylinder.

The Surface Area of Cylinder Derivation:

Suppose, we need to cover the surface of a cylinder using paper. The minimum area of paper needed to cover the curved surface of the cylinder will be the curved surface area of the cylinder. The minimum area of paper needed to cover all the surfaces (curved surface, top and bottom circular part) will be the total surface area of the cylinder. 

Let ‘r’ and  ‘h’ be the radius and height of the cylinder respectively. To cover the curved surface, a rectangular sheet of paper is needed (see the diagram below). The breadth of rectangular paper will be ‘h’ and its length will be the circumference of the circular part i.e 2πr.

Curved Surface Area (CSA) = Area of rectangular sheet

                                                 = length x breadth

                                                 = 2πr x h

                                                 = 2πrh

Total Surface Area (TSA)    = Curved Surface Area(CSA) + Area of 2 circular part

                                               =  2πrh + 2 x (πr²)

                                             = 2πrh + 2πr²

                                             = 2πr(h + r)

Sphere:

The volume of the sphere depends on the diameter of the radius of the sphere since if we take the cross-section of the sphere, it is a circle. The surface area of a sphere is the area or region which is covered by its outer surface. Formula to calculate the sphere volume, whose radius is ‘r’ is given below:

The volume of sphere = 4/3πr³cubic units.

Area of a sphere or total surface area of a sphere is the region covered by a surface of a spherical object in a three-dimensional space. Since the sphere is a complete curved shape therefore the curved surface area is equal to the total area of a sphere. It is also called a lateral surface area. 

 Area of sphere = 4πr² square unit

Comparison Between Volume of Cone and Volume of Cylinder

The volume formulas for cone and cylinder are almost similar. Let’s compare the volume of a cone and cylinder.

The volume of a cylinder is πr²h

The volume of a cone is ⅓πr²h 

Hence the volume of a cone is exactly one-third (⅓) of Volume of the cylinder.