Frustum of a Cone Volume

The word “Frustum” is originally a Latin word and the actual meaning of this word is “piece cut off”. In a frustum of a cone the upper part of the cone remains the same and with its shape but the bottom part of the cone makes a “Frustum”. To get a proper frustum, we have to slice it parallel or horizontally to the base. Both slices have different areas and volumes.

Frustum of a Cone

A Frustum is prescribed for pyramids and any type of cones. When a solid cone or generally a pyramid is cut into pieces then the base of the solid and the base of the cut slice are both parallel to each other. The solid part of the cone remains in between the cutting plain part and the surface part of the cone. To look at a frustum properly, ice cream is a perfect example to visualize the frustum accurately. When the bottom of the filled ice cream cones is cut into two slides then the section which if left behind is parallel with the base of the ice cream cone. A frustum is also defined as the headless or truncated shape. So, the other name of this type of cone is “Truncated cone”. Basically, the frustum of a cone looks like a 3D shape. In our daily life drinking water with glass is a very common activity for all. Thus a glass is a perfect example of a frustum of a cone. The glass resembles a frustum, because the smaller end of the glass is parallel with the base of the glass.

What is Frustum of a Cone Volume?

The volume of a cone is the middle solid space which is occupied by the cone itself. The two parts of a cone frustum is the top cut part and the base part of the cone. Some properties of the frustum of a cone volume are-

• The frustum of a cone does not have a vertex.
• The frustum of a cone is defined by its height and radius.
• The height of the frustum has a perpendicular distance between the two bases and the centers of the cone.
• If a cone is a right circular cone then the frustum that is made from that cone would also be a right circular frustum.

Thus, the volume of a frustum is the space inside the cone. The volume of frustum is measured by some cubic units, such as cm3, m3, in3 etc. When a three- dimensional shape with its apex or vertex cut into two parts, then the base of the cone is called a frustum. Some different types of frustum are truncated pyramid, square pyramid, triangular pyramid etc. A frustum depends on its height, its base radius 1 and base radius 2.

Volume of a frustum of a cone formula

The volume of a frustum can be calculated with some formulas. The volume of any shape depends on the height and the area of the bases of that shape. Let’s consider a frustum with its height, which is considered as ‘H’ and the two base areas, which are considered as the ‘S₁’ and ‘S₂’. So, the formula of the volume of frustum of a cone is-

“V= H/3(S₁+S₂√S₁S₂)”,

Here, H= the height of the frustum

S1= the area of any one base of that frustum

S2= another base of that frustum.

This formula can be used to calculate the volume of any frustum of a cone. Thus, here one can learn and see how to calculate the volume of a cone frustum through this formula. The other formula of the frustum of a cone is- “ πh/3 [ (R₃– r3) / r ] (OR) πH/3 (R₂+ Rr + r₂)”. Here the π is a constant and the value of the π is “22/7 (or) 3.141592653…”

Conclusion

The frustum of a cone has no vertex when it is divided into two slices; the top part of the cone remains parallel with its base. Each frustum of a cone has its own volume and surface area. The net of any frustum shape is combined by two dimensional shapes that are obtained by the three dimensional shapes. The net of a frustum cone has two circles with the two circular bases. Thus, in the geometry a frustum is a solid portion of a cone shaped figure. In computer graphics the frustum is a three- dimensional figure and the properties of a cone is convex. In this part of the assignment a huge discussion on the frustum and about the volume and formula of frustum has been elaborately discussed.