A frustum can be constructed from a circular cone in a right angle. It has a cone tip formed by cutting height which is perpendicular, an upper base and lower base. These bases in a derivation of frustum are both circular and parallel in structure.
When considering a right cone in a circular shape, the problem can be comprehensive to other n-sided pyramids and cones. Let’s take an example to understand the structure of a frustum and its application. This will further help to gain an understanding of the function of the volume of a frustum.
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In the above figure, h represents height while r is the radius of an upper base, and R is the radius. One has to apply a frustum formula to derive the volume here.
When a combination is formed by taking solids, one has to add volumes of two adjacent shapes. This will give the required volume of a structure or volume of frustum of a cone. In the case of a frustum, a cone will be cut into smaller cone ends. To find the value, one needs to subtract this separated part.
This segment explains this concept of finding volume and surface area of frustum with examples and theory.
Let’s check what the volume of frustum formula is and how it works.
How to Apply Volume of Frustum of Cone Formula?
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The sliced part of a cone can be termed as a frustum. Therefore, the calculation of volume requires finding two circular cone’s volume differences.
From the figure above, the total height becomes H’ = H+h
Here the total slant height becomes L =l1 +l2
We know that the radius of the cone is C and the radius of the cut cone is r.
This makes the volume of the total cone to be 1/3 π C2 H’ which is equal to 1/3 π C2 (H+h)
The volume of the tip of a cone here will be 1/3 πr2h. Now to find the volume of a frustum, one has to calculate the dissimilarity between two circular cones in a right angle.
This gives 1/3 π C2 H’ -1/3 πr2h
Following the rules, it gives us 1/3π C2 (H+h) -1/3 πr2h
We find that 1/3 π [ C2 (H+h)-r2 h ].
After seeing the cone, a student can assess the sliced part, which gives us a result that right angle of the whole cone Δ BAD is similar to that of sliced cone Δ BPQ.
This gives us, C/ r = H+h / h.
That is H+h = Ch/r. Replacing the value of H+h in the frustum of a cone formula we get
1/3 π [ C2 (Ch/r)-r2 h ] =1/3 π [C3h/r-r2 h]
1/3 π h (C3/r-r2 ) =1/3 π h (C3-r3 / r)
Similar Property of Triangles to Find Derivation of Volume of Frustum
If we use a similar diagram and properties, we can evaluate the value of h, C/ r to be equal to (H+h)/ h.
We have seen that here h is [r/(C-r)] H
Replacing the value of h in this equation gives us the solution 1/3 πH [r/(C-r)][(C3-r3)/ r)\]
Now we get 1/3 πH [(C3-r3)/(R-r)]
Which gives us 1/πH [(C-r)(C2 +Cr+r2)/ (C-r) ]
Finally, the value as 1/πH (C2 +Cr+r2)
Consequently, the V or the conical frustum volume will be 1/3 πH (C2 +Cr+r2 ).
How to Find Total Surface Area and Curved Surface Area in a Volume Truncated Cone
In the figure above one can find the curved surface area of the frustum of a cone to be π(C+r)l1
Here the total surface area of the frustum of a cone will be π l1 (C+r) +πC2 +πr2
We take the slant height to be l1 in both surface area of a cone. This gives us √ [H2 +(C-r)2
The resemblance of triangles equations characteristics has been calculated using two Δ BAD and Δ BPQ.
Therefore, students need to procure information on all formulas of frustum to solve equations confidently. If they practice from quality study materials, they will be accustomed to the surface area, the volume of a frustum measurement, the volume of truncated cone derivation and more.
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