Cone Formula: Definition, Surface Area, Volume & Examples

Cone is a three-dimensional shape that has a circular base and a pointed edge at the top called the apex. A cone can be thought of as a triangle that is rotated about one of its vertices. There are two types of cones namely the right circular cone and the oblique cone. Common examples of cones include ice cream cones, party hats, and traffic cones.

Cone Formula refers to a predefined set of formulas for the calculation of the surface area and volume of a cone. The curved surface area of cone is πrl whereas the total surface area of cone is πr(l + r), where r is the radius, and l refers to the slant height. The volume of cone is ⅓ πr²h, where h is the height of the cone and r is the radius.

Read More: NCERT Solutions for Class 10 Mathematics Surface Areas and Volumes

Key Terms: Cone, Cone Formula, Curved Surface Area of Cone, Volume of Cone, Surface Area of Cone, Slant Height, Radius

---

Cone Definition

[Click Here for Sample Questions]

Cone is a three-dimensional shape in Geometry that has a circular base where a set of line segments, connect all of the points on the base to a common point called the apex or the vertex. A cone has one face, one vertex, and no edges. Examples of cones are birthday caps, a tent, ice cream cones, etc. 

There are three elements of a cone which are its radius, height, and slant height. 

• Radius (r) is defined as the distance between the center of the circular base to any point on the boundary of the base. 
• Height (h) of the cone is the distance between the apex of the cone to the center of the circular base. 
• Slant Height (l) is the distance between the apex of the cone to any point on the circumference of the cone. It is calculated as l² = r² + h².

Cone

Read More:

---

Types of Cone

[Click Here for Sample Questions]

There are two main types of cones in Geometry: 

1. Right Circular Cone
2. Oblique Cone

Right Circular Cone

The right circular cone is the most common type of geometric cone. 

• It is a type of cone that has a line that touches the top point of the cone in a perpendicular. 
• In this, the line passes through the center of the circle. 
• The apex of the cone lies just above the center of a circular base. 
• Examples of the right circular cones are ice cream, traffic cones, etc.

Oblique Cone

The oblique cone is the other type of cone in geometry that is tilted or slanted.

• The base and the apex of the oblique cone are not perpendicular to each other. 
• The top point of the cone is not perpendicular to the base. 
• In this cone, the vertex of the cone does not lie above the centre of the circular base. 
• An example of an oblique cone is a geometrical conical figure.

Read More: Surface Areas and Volumes Important Questions 

---

Cone Formula

[Click Here for Sample Questions]

Cone formula refers to a set of formulas that are used to calculate the surface area and volume of a cone. The various cone formulas are 

1. Curved Surface Area of Cone Formula
2. Total Surface Area of Cone Formula
3. Volume of Cone Formula

These are the three formulas that are used to calculate the area, volume, and various dimensions of a given cone.

---

Curved Surface Area of Cone

[Click Here for Sample Questions]

The curved surface area of cone refers to the area enclosed by the curved part of the cone. 

Curved Surface Area of Cone = πrl square units

Where 

• r = Radius of Cone
• l = Slant Height of Cone

Derivation of CSA of Cone Formula

To derive the curved surface area of a cone, we will divide the given cone into a circular base and the top slanted part. The area of the slanted part gives us the curved surface area. 

CSA of Cone Formula

Area of the Circular Base: The area of the circular base is given as

Area of a circle = πr²

If we open the curved top and cut it into small pieces so that each cut portion is a small triangle, whose height is the slant height l of the cone.

Area of the Circular Base

So, the area of each triangle = 1/2× base of each triangle × l.

Therefore, 

Area of the curved surface = Sum of Areas of all the Triangles

=½ × b₁ × l + ½ × b₂ × l+ ½ × b₃ × l+ ………+ ½ × b_n × 1

=1/2l (b₁ + b₂ + b₃ +……+b_n)

=1/2l

Lastly, 

Length of Entire curved Boundary = Circumference of base = 2πr 

Thus,

Area of the curved surface = ½ × l × 2πr = πrl

Hence derived. 

Thus, the curved surface area of a cone is calculated as πrl.

Read More: MCQs On Surface Area And Volume

---

Total Surface Area of Cone

[Click Here for Sample Questions]

The total surface area of a cone is the sum of the area of the circular base and the area of the curved part of the cone. 

Total Surface Area (TSA) of Cone = Area of the Base (Circle) + Curved Surface Area of the Cone (CSA)

TSA = πr² + πrl = πr (l + r) square units

Total surface area is sometimes referred to as only the surface area of a cone. 

Derivation of TSA of Cone Formula

According to the definition of the total surface area of a cone, 

Total Surface Area of a Cone = Area of the Circular Base + CSA of Cone

Total Surface Area of a Cone = πr² + πrl = πr (l + r)

Thus, the total surface area of a cone = πr (l + r)

---

Volume of Cone

[Click Here for Sample Questions]

The volume of a cone refers to the space occupied by the cone. 

Volume of Cone = (1/3) πr²h cubic units

Where 

• π = Value of pi.
• r = Radius of the circular base.
• h = Height of the cone.

The volume of a cone is also one-third of the volume of a cylinder.

Volume of Cone = (1/3) × Volume of Cylinder

Read More:

Solved Examples on Cone Formula

Example 1: What will be the curved surface of a cone having a base radius of 5 cm and a slant height of 20 cm? (π = 22/7)

Solution: Given parameters are,

• Radius (r) = 8 cm
• Slant height (l) = 20 cm

Using the curved surface area of a cone formula,

CSA of Cone = πrl

CSA of Cone = 22/7 × 5 × 20 = 314.08 cm²

Thus, the curved surface area of the given cone is 314.08 cm².

Example 2: Sheetal observes a conical tent during her camping trip and approximates that the height of the tent is three times the radius of the tent which is given as 3 m. Calculate what will be the volume of the tent.

Solution: According to the question, the height of the tent is three times the radius. 

So, 

h = 3r

h = 3 x 3 = 9 m

Volume of Cone = (1/3) πr²h

Substituting the value of h = 9, we get,

Volume of Conical Tent = (1/3) πr² × 9

= (1/3) x 22/7 x 3 x 3 x 9

= 84.82 cm³

Therefore, the volume of the tent is 84.82 cm³.

Example 3: There is a conical birthday cap with a radius of 3 units and a height of 4 units. What will be the slant height of the birthday cap?

Solution: Given that,

• Radius (r) = 3 units 
• Height (h) = 4 units

Using slant height of a cone formula, 

l² = r² + h²

3² + 4² = 9 + 16

l² = 25

l = √25

l = 5

Thus, the slant height of the birthday cap is 5 units.

---

Things to Remember

• Cone is a 3D shape with a circular base and a pointed edge at the top called the apex.
• There is 1 face, 1 vertex, and no edges in a cone.
• The right circular cone and the oblique cone are the two types of cone. 
• The total surface area of a cone is πr (l + r) square units.
• The curved surface of a cone is πrl square units.
• The volume of a cone is given as (1/3) πr²h cubic units.
• The slant height of a cone can be calculated using the formula l² = r² + h².