Frustum
A frustum is a unique 3D object that is derived by cutting the apex of a cone or a pyramid. The portion that is left with us after cutting the apex of a cone or a pyramid is known as a frustum. The top and bottom bases of a frustum are parallel to each other.
| 1. | Definition of Frustum |
| 2. | Types of a Frustum |
| 3. | Properties of a Frustum |
| 4. | Volume of a Frustum |
| 5. | Surface Area of a Frustum |
| 6. | Lateral Surface Area of a Frustum |
| 7. | Net of a frustum |
| 8. | FAQs |
Definition of Frustum
A frustum can be defined as a solid shape obtained from cutting a cone or a pyramid from top. It is that section of the cone or pyramid which lies between the base and the plane parallel to the base. It is a 3D figure obtained from other 3D 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 of a frustum for your reference.
Types of Frustum
A frustum is a 3D shape enclosed between the two parallel planes of another solid that is cut into two parts. There are generally two types of frustum:
- A cone frustum: Created by cutting the cone from the vertex or apex. A plane parallel to the base of the cone cuts the top of the cone or the apex to create a frustum. It is also called a frustum of a cone or truncated cone.
- A pyramid frustum: Formed 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. As a geometrical figure, it has many properties that distinguish it from other solid shapes. The properties of the frustum are listed below:
- A frustum can be determined by the height of the solid figure remained after cutting off a cone or pyramid.
- Its radius can be determined by calculating the radius of both the bases 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 (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 square-shaped 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 cm3, m3, in3, etc. The volume of a frustum 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), S1 is the area of one base, and S2 is the area of the other base. 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 above volume formula as well, but when the questions are more specific on mentioning that it is cone, we can use the following formula:
Volume of frustum of cone = πH/3 (R2 + Rr + r2)
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 a 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) + πR2 + πr2 square units, where π is a constant whose value is 22/7 (or) 3.142 approx, R is the radius of the larger base, r is the radius of the smaller 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 circumference 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 its base, 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 one is bigger in size. Here is an image of the frustum net for your reference:
Related Topics on Frustum
Listed below are a few interesting topics related to the concept of the frustum. Have a look!
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. It can be a cone-based (obtained from a cone), triangle-based (obtained from a triangular pyramid), and a square-based frustum (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 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: It is obtained by cutting the cone from the top. It is also called a 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 a square.
What is the Formula to Find the Volume of a Frustum?
The volume of a frustum can be determined by using its height and the area of its bases. Hence, the formula to calculate the volume of the frustum is \(\dfrac{H}{3}\left(S_{1}+S_{2}+\sqrt{S_{1} S_{2}}\right)\) cubic units, where H is the height of the frustum (the distance between the centers of two bases of the frustum), S1 is the area of one base, and S2 is the area of the other base of the frustum.
How 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. Therefore, the formula to find the volume of the frustum of a cone is πH/3 (R2 + Rr + r2) cubic units, where π is a constant whose value is 22/7 (or) 3.142 approx, H is the height of frustum, 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 πL (R + r) + πR2 + πr2 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 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).