Magical Education

Percentage loss is a crucial concept. Real life examples, finance and statistics are directly or indirectly based on the concept of percentage loss.

Now, we will be substituting this value in the equation1 so that we can have the other equation as:The frustum of a cone is defined as that part of cone that it is cut by a plane into two parts.  This is the upper part of the one remains that in the shape in this case the lower part will make the frustum. In order to get to this part of the right circular cone we must slice it horizontally or parallel to the base.  The figure will be taken into consideration

The term Frustum is a Latin word which means ‘piece cut off’. The solid structure that is generally a part of the cone or pyramid that is cut in a manner that base of the solid and that of the plane cutting the solid will be parallel to each other.  This part remain between the parallel cutting plane and the base is called the frustum of that solid.

consider an ice-cream cone which is filled with ice-cream. So, if we cut the cone is in a manner as being depicted in the figure below, the section that will be left between the base and parallel plane is called the frustum of a cone.

VOLUME OF FRUSTUM OF THE CONE

The volume of frustum of a cone with the help of diagram that is being given below. The application of the formula will help to find the volume of frustum of a pyramid that is shaped in the form of a cone.  The structure is explained as:

This will take the larger cone which has a height equal to h units; let us assume slant height as l units and radius as r units for the first cone 1. On the same grounds, the smaller right circular cone be named as cone 2 and here we will take   h′ units, radius as r′ units and the slant height as units.

The height of frustum is H units and its slant height is L units.

The volume of right circular cone 1 = 1/3 πr2h  

Similarly, the volume of right circular cone 2 = 1/3 πr′2h′

So, here have the calculate the process with the implementation of the formula as well so that the volumes can also be calculated here.

the volume of the frustum of cone can be given as:

From the figure given above, in the ∆OO′D and ∆OPB.

Applying the condition that is applied for the similar triangle so that this will be able to have the application of the same principle in the corresponding mechanism as well. Deductions are:

Now, we will be substituting this value in the equation1 so that we can have the other equation as:

So, as we can see that the substitution can also be done in the figure 3, so it can be written as:

h= H+h’

substituting the final value in the equation 2 so that this will compute the value:

Lastly, we will be substituting the value of h in the final equation of 3 so that we will be able to compute the value as:

Surface Area of Frustum of Cone

The surface area will also be the difference of the surface areas of both the cones in the pattern. After the deduction we can have the parameters be defined as:

the calculated curved surface area of right circular cone 1 is = πrl

 Curved surface area of right circular of cone 2 = πr’l’

 Hence, Curved surface area of the frustum of cone will be calculated as = πrl – πr’l’

So, when we deduce it from fig. 3, in figure the ∆OAB and ∆OCD.

∠AOB = ∠COD (Common Angle)

 This is being defined as the Plane dividing the cone is parallel to the base.

⇒ ∠OCD = ∠OAB (is defined as the corresponding Angles)

Thus, ΔOAB~ΔOCD by the application of the Angle similarity.

So, if we calculate the lateral surface area of frustum of the cone. It is defined as the difference of the areas of sector of circles (s and s′) with radii r and r′ and that of the common central angle θ that is being shown in the diagram.

In this case if we must apply the initial calculations and the substitution of the vales with the process mechanism, we will be able to arrive at the formula for the surface area of frustum defined as