Frustum of a Cone – Definition, Surface Area & Volume Formulas Explained

A frustum of a cone is formed when a cone is cut by a flat surface (plane) that is parallel to its base. The part of the cone between this cut and the base is called the frustum. The word "frustum" comes from Latin and means "a piece cut off." In geometry, a cone is a 3D shape with a circular base and a pointed top called the apex. When the top portion of a cone is removed in a straight, flat cut, the remaining bottom part becomes a frustum. The shape still has two circular faces – one large and one smaller – but no apex. Frustums are commonly seen in everyday life. Some examples include a bucket, a flower pot, a glass tumbler, or a lampshade. These objects all have a wider base and a narrower top, just like the frustum of a cone.

What is Frustum of a Cone?

A frustum is the part of a solid shape that lies between two parallel flat surfaces that cut through it. When we cut a cone with a plane that is parallel to its circular base, the portion of the cone between the base and the cut is called the frustum of a cone. It has two circular faces – one bigger and one smaller – and a curved surface connecting them. The pointed top of the cone is removed, leaving the flat top instead. Buckets and lampshades are common examples of frustums in real life.

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The frustum of a cone is also known as the truncated cone. Just like any other 3D shape, it also has surface area and volume. Some well-known real-life examples of it are the shade of a table lamp, bucket, glass tumbler, etc.

Let us now define some terms related to the frustum of a cone such as height, slant height and radius.

Height: The height or thickness of the frustum is the perpendicular distance between its two circular bases. It is generally denoted by ‘’.

Slant Height: The slant height of a frustum of a cone is the length of the line segment joining the extremities of two parallel radii, drawn in the same direction of the two circular bases. It is generally denoted by ‘’.

Radii: As a frustum is having two circular bases, there will be two radii in a frustum. It is generally denoted by ‘’ and ‘’.

Frustum of a Cone Formulas

Consider height to be ‘’, slant height to be ‘’, ‘’ and ‘’ be the two radii of circular bases of the frustum respectively. Then the formula related to the frustum of a cone is tabulated below:

Frustum of a Cone	Formula
Slant Height of Frustum of a Cone
Curved Surface Area (CSA) or Lateral Surface Area (LSA)	CSA or LSA
Total Surface Area (TSA)	TSA
Volume

Volume of Frustum of a Cone

The volume of frustum of a cone is the volume of the solid shape obtained when a right-circular cone is cut by a plane, parallel to the base. It depends on its slant height and radius of the upper and bottom circular part.

Volume of frustum of a cone is

Derivation of Volume of Frustum of a Cone Formula

Let ‘’ and ‘’ be the height of cone and frustum, ‘’ and ‘’ be the slant height of the cone and frustum respectively.

If ‘’, ‘’ are the radii of the circular bases of the frustum, then volume of the frustum of a cone is the difference between the volumes of the two cones, i.e.

Since the triangles ABC and ADE are similar, then the ratio of their corresponding sides is proportional.

……..(i)

[using eq.(i)]

Therefore, the volume of the frustum of a cone is cubic units.

Surface Area of Frustum of a Cone

The surface area of the frustum of a cone is basically the sum of the areas of its curved surface and its two circular faces. It is measured in square units. There are two types of surface area of the frustum of a cone:

• Curved surface area (CSA) or Lateral surface area (LSA)
• Total surface area (TSA)

Curved Surface Area of Frustum of a Cone

Curved surface area or lateral surface area of a frustum of a cone is the area of its curved surface. Curved surface area (CSA) or lateral surface area (LSA) is given as CSA or LSA

Derivation of Curved Surface Area of Frustum of a Cone Formula

Let ‘’, ‘’, and ‘’ be the height of the cone, sliced cone, and frustum, ‘’, ‘’, and ‘’ be the slant height of the cone, sliced cone, and frustum, ‘’, ‘’ are the radii of the circular bases of the frustum respectively.

From the figure, we have total slant height .

Since the triangles ONB and OMC are similar, then the ratio of their corresponding sides is proportional.

……..(i)

Now, Curved surface area of the frustum of a cone = Curved surface area of the original cone – Curved surface area of the sliced cone

[using eq.(i)]

[ ]

Therefore, the curved surface area of the frustum of a cone is square units.

Total Surface Area of Frustum of a Cone

The total surface area of a frustum of a cone is the sum of the areas of its curved surface and its two circular faces. Total surface area (TSA) of frustum of a cone is given as TSA

Derivation of Total Surface Area of Frustum of a Cone Formula

Let ‘’, ‘’ are the radii of the circular bases of the frustum respectively.

Now, Total surface area of the frustum = Curved surface area of the frustum + Area of upper circular base + Area of lower circular base

Since, Curved surface area of the frustum of a cone = ,

Area of upper circular base (radius is ‘’) = , and

Area of lower circular base (radius is ‘’) = .

Then,

Therefore, the total surface area of the frustum of a cone is square units.

Slant Height of Frustum of a Cone

The slant height of a frustum of a right circular cone is the length of the line segment joining the extremities of two parallel radii, drawn in the same direction of the two circular bases. It is generally denoted by ‘’. Slant height of frustum of a cone is given as:

Derivation of Slant Height of Frustum of a Cone Formula

Let ‘’ be the height of frustum, ‘’ be the slant height of the cone, ‘’, ‘’ are the radii of the circular bases of the frustum respectively.

In , . By Pythagoras Theorem, we have

Taking square root on both sides, we get

Therefore, the slant height of the frustum of a cone is .

Properties of Frustum of a Cone

The properties of a frustum of a cone are given below:

• The height of a frustum of a cone is the perpendicular distance between the two bases of the frustum
• It doesn’t contain the vertex of the corresponding cone but contains the base of the cone.
• It is determined by its height and two radii (corresponding to two bases).
• If the cone is a right-circular cone, then the frustums formed from it also would be right-circular.

Net of Frustum of a Cone

The net of any form is formed up of several two-dimensional shapes produced by releasing three-dimensional shapes. In other terms, whenever the net of a frustum gets folded up, the matching frustum is made. The net of a frustum of a cone has two circles corresponding to its two circular bases.

Ratio of Volume of Cone to Volume of Frustum

Let be the volume of a cone and be the volume of the frustum of a cone respectively.

, where be the radius, be the height of the cone.

, where is the height, and are the two circular radii of the frustum of a cone.

Then, the ratio of volume of a cone to volume of a frustum is given by

.

Ratio of Surface Area of a Cone to Frustum

Let be the surface area of a cone and be the surface area of the frustum of a cone respectively.

Case 1: Ratio of curved surface area of a cone to frustum:

Here , where is the radius and is the slant height of the frustum.

, where is the slant height of the frustum, and is the two circular radii of the frustum.

Then, the ratio of curved surface area of a cone to curved surface area of a frustum is given by

.

Case 2: Ratio of total surface area of a cone to frustum:

Here , where be the radius, be the slant height of the cone.

, where be the slant height of the frustum of a cone, and is the two circular radii of the frustum of a cone.

Then, the ratio of curved surface area of a cone to curved surface area of a frustum is given by

.

Important Notes

• π (pi) is a special constant used in geometry. Its value is either 22/7 or approximately 3.14159.
• For a frustum of a cone, the slant height (L), vertical height (H), and the radii of the two circular bases (R and r) are related by the Pythagoras Theorem:
  L² = H² + (R - r)²
• This formula is useful when the problem gives you the slant height instead of the vertical height, or when you need to find one of them.

Solved Examples of Frustum of a Cone

Example 1: If the radii of the circular ends of a conical bucket which is cm high are cm and cm. Find the capacity of the bucket. Use .

Solution: The bucket forms a frustum of cone such that radii of its circular ends are cm and cm, and the height is cm.

Let V be the capacity or the volume of the bucket.

Then, the volume of the frustum of the cone is,

[latexV=15400[/latex]

Therefore, the capacity of the given bucket is cm.

Example 2: Find the total surface area of the frustum of a right circular cone of height in, large base radius to be in, and slant height to be in. Express the answer in terms of .

Solution: The larger base radius of the frustum of a cone is, in.

Let the smaller base radius be ‘’.

The height of the frustum of the cone is, in.

The slant height of the frustum of the cone is, in.

We know that,

Squaring both sides, we get

Taking square root on both sides, we get

The total surface area of the given frustum of a right circular cone is,

TSA

TSA

TSA

TSA

Therefore, the frustum of the right circular cone is in.

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If you are checking Frustum of a Cone article, check the related maths articles:
Volume of Sphere	Volume of Cylinder
Volume of a Frustum	Volume of a Pyramid
Volume of a Cuboid	Volume of a Cube

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