Frustum
A frustum is a unique object that is derived by cutting the apex of a cone or a pyramid into a flat object from top and bottom. The frustum can be a part of any solid object between two parallel lines. Once the frustum is derived from the cutting of the figures, we can say the top and bottom of a frustum are parallel to each other. Let's learn more about this interesting concept and solve a few examples to understand it better.
Definition of Frustum
A frustum can be defined as a solid shape obtained from cutting a cone or a pyramid from top and bottom between two parallel planes or lines. It is that section of the cone or pyramid which lies between the base and the plane parallel to the base. It is a 3Dshaped solid figure obtained from other 3D shape figures such as a cone or pyramid. A frustum is determined by its height, its radius of base 1 (top), the radius of base 2 (bottom). Below is an image for your reference.
Types of Frustum
A frustum is a 3D shape that belongs between the two parallel planes of another solid that is cut into two parts. The cut section between the two planes is the frustum. There are two general types of frustum:
 A cone frustum: Created by cutting the cone from the top. A plane parallel to the base of the cone cuts the top of the cone or the apex to create a frustum with a base on top as well. It is also called a frustum of a cone or truncated cone.
 A pyramid frustum: Created by cutting the apex of the pyramid with a plane parallel to the base. Here, the pyramid's base can either be a triangle or a square. Hence, frustum can be created from triangular pyramids and square pyramids.
Properties of Frustum
The word frustum is a Latin word that means piece cut off. When a cone or a pyramid is cut off from the top or apex with a plane parallel to the base of a cone or pyramid, the solid figure that remains is called a frustum. This solid has various properties, let's take a look at them:
 A frustum can be determined by the height of the solid figure remaining after cutting off a cone or pyramid
 The radius of a frustum can be determined by calculating the radius of both the base i.e. the top and bottom
 The plane part of the frustum is called the floor
 If we find an axis of a frustum, it is mostly the axis of the cone or pyramid which was the original shape
 If the axis is perpendicular to the base, then the frustum is a right frustum otherwise it is an oblique frustum
 A frustum can have a triangle and squareshaped base if a triangular pyramid and a square pyramid are cut off from the apex.
Volume of Frustum
The amount of space that is present inside a frustum is called the volume of a frustum. The volume helps in calculating how much matter can be stored or how much a frustum can hold inside it. It is measured in cubic units such as cm^{3}, m^{3}, in^{3}, etc. The volume of a frustum of any shape can be determined by using its height and the area of its bases. Hence, the formula to calculate the frustum volume is:
Volume of frustum = \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\)
where H is the height of the frustum (the distance between the centers of two bases of the frustum), S_{1} is the area of one base of the frustum, and S_{2} is the area of the other base of the frustum. You can learn the method of finding the volume of a frustum by reading an interesting article on Volume of Frustum.
However, in geometry, we usually come across questions on calculating the volume of the frustum of a cone. We can use the previous volume as well but when the questions are more specific on mentioning that it is cone, we can use the following formula. Most of the time this is the formula that is used.
Volume of frustum of cone = πH/3 (R^{2} + Rr + r^{2})
Where π is a constant whose value is 22/7 (or) 3.142 approx, R is the radius of the bottom base, and r is the radius of the top base. You can read all about it by checking out this page Frustum of a right cone formula
Surface Area of Frustum
The surface area of a frustum can be determined by calculating the area of the bases and the slant height. To do that, here is the formula to calculate the surface area of a frustum made from a cone in most cases.
Surface Area of Frustum of Cone = πL (R + r) + πR^{2} + πr^{2} square units, where π is a constant whose value is 22/7 (or) 3.142 approx, R is the radius of the bottom base, r is the radius of the top base, and L is the slant height.
Lateral Surface Area of Frustum
The lateral surface area or the curved surface area of a frustum of a cone can be determined by calculating the difference of the areas of the sector of circles i.e. top and bottom base, with a common central angle. The forumla to calculate the lateral surface area is:
Lateral Surface Area of Frustum of Cone = πL(R+r) square units, where π is a constant whose value is 22/7 (or) 3.142 approx, R is the radius of the bottom base, r is the radius of the top base, and L is the slant height.
For more information on both surface area and lateral area of the frustum, check out this interesting article.
Net of a Frustum
When a pyramid or a cone is cut off by using a plane that is parallel to the base of either a pyramid or cone, the solid structure left behind is called a frustum. The net of a frustum is formed by a shape similar to a cylinder and two circles  one small in size and the other bigger in size. When a frustum is opened up we can see the shape of half a cone which is curved or the shape of half a pyramid with the base either a square or a triangle. Here is an image of the frustum for your reference:
Related Topics on Frustum
Listed below are a few interesting topics related to the concept of the frustum. Have a look!
Solved Examples of a Frustum

Example 1: The bases of a frustum of a square pyramid are of lengths 12 units and 8 units. Its height is 14 units. Find the volume of the frustum.
Solution:
Given,
The height of the frustum of the square pyramid, H is 14 units.
The areas of bases of the frustum are: S1= 12^{2} = 144 square units and S_{2}= 8^{2} = 64 square unit
The volume of the frustum of the square pyramid is,
Volume of frustum = \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\)
Volume = 14/3[144 + 64 + √(144×64)]
Volume = 14/3(208 + 96)
Volume = 1418.6 square units
Therefore, the volume of the frustum of the given square pyramid is 1418.6 square units.

Example 2: Find the volume of the frustum of a cone of height 10 in, large base radius 15 in, and small base radius is 5 in. Express the answer in terms of π.
Solution:
Given, H = 10, R = 15, r = 5
Let us put the values in the formula, Volume of frustum of cone = πH/3 (R^{2} + Rr + r^{2})
Volume = π(10/3)(15^{2} + 15 × 5 + 5^{2})
Volume = π(10/3)(325)
Volume = 1083.4 π
Therefore, the volume of a frustum of a cone is 1083.4 π cubic inches.
FAQs on Frustum
What is a Frustum?
A frustum is a solid 3D object that is obtained by cutting the top or apex of a cone or pyramid with a plane that is parallel to the base of the cone or pyramid. The solid that is left behind after cutting the top horizontally is the frustum. A frustum can be a conebased obtained from a cone, can be a trianglebased obtained from a triangular pyramid, and can be a squarebased obtained from a square pyramid.
What are the Types of Frustum?
A frustum is of two kinds  frustum of a cone and frustum of a pyramid. In geometry, the frustum of a cone is usually used most often and the volume, surface area, and lateral area are usually based on the frustum of the cone. The types are:
 A cone frustum: This is obtained by cutting the cone from the top. It is also called a frustum of a cone or truncated cone
 A pyramid frustum: It is obtained by cutting the apex of the pyramid with a plane parallel to the base that can either be a triangle or square
What is the Formula to Find the Volume of a Frustum?
The volume of a frustum of any shape can be determined by using its height and the area of its bases. Hence, the formula to calculate the volume of the frustum is Volume of frustum = \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\) cubic units, where H is the height of the frustum (the distance between the centers of two bases of the frustum), S_{1} is the area of one base, and S_{2} is the area of the other base of the frustum.
What is the Formula to Find the Volume of Frustum of Cone?
In geometry, most of the time the word frustum is usually indicated as a frustum of a cone. Hence, the formula is a little different from the general formula as mentioned in the previous question. Therefore, the formula to find the volume of the frustum of a cone is, Volume of frustum of cone = πH/3 (R^{2} + Rr + r^{2}) cubic units, where π is a constant whose value is 22/7 (or) 3.142 approx, R is the radius of the bottom base, and r is the radius of the top base.
What is the Formula to Find the Surface Area of a Frustum of Cone?
The surface area to find the volume of the frustum of a cone is, Surface Area of Frustum of Cone = πL (R + r) + πR^{2} + πr^{2} square units, where π is a constant whose value is 22/7 (or) 3.142, R is the radius of the bottom base, r is the radius of the top base, and L is the slant height.
What is the Formula to Find the Lateral Surface Area of Frustum of Cone?
The formula to find the lateral surface area of a frustum of a cone is, Lateral Surface Area of Frustum of Cone = πL(R+r) square units, where π is a constant whose value is 22/7 (or) 3.142, R is the radius of the bottom base, r is the radius of the top base, and L is the slant height.
What is the Difference Between a Cone and Frustum?
A cone is a 3D shape with a base shaped like a circle with the top is pointed upwards. Whereas a frustum is a solid obtained by cutting the top of a cone with a plane that is parallel to the circular base. A frustum of a cone looks like a glass with two circular bases (top and bottom).