Volume of a Frustum

Dhristi JEE 2022-24

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.

One can check Vedantu to gather quality guidance on the topic volume of a frustum of a pyramid and related topics. They offer a wide array of notes, practice papers, live classes, and more. Student’s desiring to secure outstanding scores in the subject maths can download the app today!

FAQs on Volume of a Frustum

1. What will be the Cost of Oil When an Open Drum Has Height 63 cm with a Radius of Lower and Upper Ends as 15 cm and 25 cm. It Can Fill the Bucket at Rs. 42 Per Litre.

Ans.  A plastic drum looks like a frustum of a cone where the Height h is 63 cm. We have Upper Radius (R) as 20 cm of the drum. 

Here the Lower Radius r is given as 15 cm. So, using a formula for the volume of the frustum of the cone, one will find the volume to be 1/3πH (R2 + Rr + r2)

This gives us 1/3 × 22/7 × 63(20² + 20×15 +15²)

Now, 66 (400 + 300 + 225)

Then, 61,050 cm3

Finally, we get the value as 61 L 50 ml

One litre of oil costs Rs 42, and 61 L & 050 ml will cost around 61.050×42 

This makes a total of Rs 2564.1 

The Drum stores oil of cost = Rs 2564.1 

Total Surface Area of the Drum will be π l (R+r) + πR2 + πr2

l =√H2 +(R-r)2

l = 632 + (20-15) 2

Which makes √3969+25

The total area will be 88cm.

2. What Do You Mean By Convex Polytope in a Cone?

Ans. A convex polytope is a form of polytope that has supplementary property. It is parallel to a convex set in a Euclidean space. Many use this term polytype to explain a convex polytope that is bounded. This term polyhedron can also be used to explain an unbound item or object. 

Many use terms like unbound or bound convex polytope to explain the importance of this shape. It can also be defined through its boundary and functions. A convex polytope holds its value in the fields like mathematics and programming for linear structure.

3. What Does the Surface Area of an Object Mean?

Ans. The surface area of an object is a measurement of an objects’ occupancy of total area. In maths, the surface area in a curved surface is more active than a polyhedral surface. It is also more involved than a one-dimensional arc length of a curve which makes the total area of faces as the value of a surface area. 

One can find parametric surfaces in an even surface of objects like spheres. Thus, the surface has the involvement of double integration, infinitesimal calculus and partial derivatives.