Right Circular Cone

A right circular cone is a 3D shape with a circular base and a curved surface that narrows to a point known as the apex or vertex. The cone’s axis is the line connecting the vertex (apex) to the centre (midpoint) of the circular base. This axis is perpendicular to the base, creating a right angle.

The volume formula for a right circular cone is V = (1/3) × πr²h, where:

• r is the radius of the base circle cone
• h is the height of the cone

This formula tells us that the volume of a cone is one-third of the volume of a cylinder with the same radius and height.

What is a Right Circular Cone?

A cone whose axis is perpendicular to the plane of the base of the cone is called a right circular cone. It is a 3-D figure in which a flat circular surface meets a curved surface at the point on top of the figure. We can imagine the right circular cone as the shape drawn out from the cylinder taking its base,, but as we start to go up, the radius of the cone gets smaller until it becomes zero,, and the cone is closed at that point.

This point is called the vertex of the cone,, and the circular,, flat surface is called the base of the cone. A triangle rotated from the smaller sides forms a cone. Now let’s learn more about the right circular cone,, its formulas, properties,, and others in detail in this article. 

Right Circular Cone Definition

If the axis in any cone is perpendicular to the plane of the base of the cone, then the cone is called a right circular cone. We can form a right circular cone by rotating any right-angle triangle along its base or height.

Parts of a Right Circular Cone

We have the following three parts to any right circular cone:

• Radius of Base
• Height of Right Circular Cone
• Slant Height of Right Circular Cone

Radius of Base: It is defined as the base radius of the circular base of the cone. It is generally denoted using ‘r‘.

Height of the Right Circular Cone: It is defined as the height of the right circular cone. It is the perpendicular distance between the centre and the vertex of the circle. It is generally denoted using ‘h‘.

Slant Height of the Right Circular Cone: It is defined as the length between the vertex and the point on the circumference of the base off the circular plane measured along the surface of the cone. It is generally denoted using ‘l‘.

The relation between the radius of the base (r), height (h), and slant height (l) of the right circular cone is:

l² = h² + r²

Right Cone vs Oblique Cone

A cone is a 3D shape that resembles a triangular shape with a circular base. We can classify the cones into two types based on their alignment of the vertex and the centre of the circular base:

• Right Circular Cone
• Oblique Cone

Right Circular Cone: If the centre of the circular base of the cone is perpendicular to the vertex or apex of the cone,, it is called the right circular cone.

Oblique Cone: If the centre of the circular base of the cone is not perpendicular to the vertex or apex of the cone,, it is called the oblique cone.

The image added below shows a right circular cone and an oblique cone.

We can also check the type of cone by taking a plane parallel to the base of the cone and intersecting it with the curved surface of the cone. If the intersection results in a circular shape, the cone is a right circular cone. If the intersection results in an oval shape, the cone is an oblique cone.

Properties of a Right Circular Cone

A right circular cone,, or simply a right cone,, is a unique 3D shape that has various properties. Some of the important properties of the right cone are:

• Axis of the right cone is a line joining the vertex to the centre of the circular base.
• Rotating a right triangle along its base or perpendicular generates the right circular cone, in which the area generated by the hypotenuse of the cone is the curved surface area of the right cone.
• A plane parallel to the base of a right circular cone if intersecting with the surface of the right cone results in a circle.
• Height of the right circular cone is the perpendicular distance between the centre of the circular base and the vertex of the cone.

Surface Area of a Right Cone

Surface area of a right circular cone is the total region occupied by the surface of the three-dimensional shape and is measured in terms of square units such as sq. cm (cm²), sq. m (m²), etc. A right circular cone has two surface areas:

• Curved Surface Area (CSA)
• Total Surface Area (TSA)

A right circular cone with radius (r), height (h), and slant height (l) is shown in the image below:

Now let’s learn about the the CSA and TSA of the right circular cone in detail.

Curved Surface Area of a Right Circular Cone

The lateral surface area,, or curved surface area,, of the right circular cone is the region occupied by the curved surface of the right circular cone. The area of the base is excluded when we calculate the curved surface area of a right circular cone. The formula to calculate the curved surface area of the right circular cone is:

CSA of a Right Circular Cone = πrl square units

We know that l² = h² + r², so the CSA of the cone is also written as:

CSA of a Right Circular Cone = πr√(h² + r²) square units

where,

• r is the Radius of Base
• l is the Slant Height of Cone
• h is the Height of Cone

Example: Find the curved surface area of a right cone if its radius is 28 units and its height is 45 units.

Solution:

Given,

Radius (r) = 28 units 

Height (h) = 45 units

We know that,

Curved Surface Area of Cone = πr√(h² + r²) square units

= (22/7) × 28  × √(28² + 45²)

= 22 × 4 × √(784 + 2025)

= 22 × 4 × √(2809)

= 22 × 4 × 53

= 4664‬ square units.

Hence, the curved surface area of the cone is 4664 square units.

Total Surface Area of a Right Circular Cone

The total surface area of the right circular cone is the total area of the right circular cone,, including the area of the circular base and the CSA of the right circular cone. The formula to calculate the total surface area (TSA) of the right circular cone is:

Total Surface Area of Right Circular Cone (TSA) =  Area of Circular Base + CSA of Right Circular Cone

Now:

• Area of Circular Base = πr²
• CSA of Right Circular Cone = πrl

TSA = πr² + πrl  

TSA = πr(r + l)

Hence,

Total Surface Area of Right Circular Cone = πr (r + l) square units

where,

• r is Radius of Base
• l is Slant Height of Cone

Example: Calculate the surface area when the radius and slant height of a right cone arecone, 10 units and 11 units, respectively. (Use π = 22/7)

Solution:

Given,

Radius (r) = 10 units

Slant Height (l) = 11 units

We know that,

TSA of right cone = πr(r + l) square units

TSA = 22/7 × 10 × (10 + 11) 

        = 22 × 30 

        = 660 sq. units

Hence, the total surface area of right circular cone is 660 sq.units

Volume of a Right Circular Cone

The total space occupied by the right circular cone is defined as the volume of the right circular cone. We can also define the volume of the right circular cone as the amount of total material that the cone contains. The volume of a cone is expressed in cubic units like m³, cm^3, etc.

Volume of a Right Circular Cone Formula

The formula used to calculate the volume of the right circular cone is discussed in the article below. Suppose we have a right circular cone with radius (r) and height (h), then its volume is 1/3 of the volume of the cylinder of the same dimension, i.e.

Volume of Right Cone = (1/3) × Volume of the cylinder

For calculating the formula for volume of the right cone, we define the volume as:

Volume of Right Cone = (1/3) × Area of Circular Base × Height of the Cone

We know that.

• Area of Circular Base = πr²
• Height of the Cone = h

V = (1/3) × πr² × h

where,

• r is Radius of Base
• h is Height of Right Circular Cone

Other Formulas

We know that the slant height of a  cone (l) = √(r² + h²)

So, by replacing the value of slant in the surface areas formula of a right cone, we get

• Curved Surface Area of a Right Cone (CSA) = πr√(r² + h²) square units
• Total Surface Area of a Right Cone (TSA) = πr² + πr√(r² + h²) square units

Frustum of a Right Circular Cone

If we cut a right circular cone by a plane parallel to its base, the figure so obtained is called the frustum of the right circular cone. There are various objects that are in the shape of the frustum of the cone. The most common object that resembles the frustum of a right circular cone is the bucketthe.

The volume of the frustum of a right circular cone is obtained by subtracting the volume of the small cone from the volume of the larger cone.

Equation of Right Circular Cone

We can represent the right circular cone in 3D space with the equation discussed below. If the vertex of the cone is at the origin and the cone is arranged in x, y, and z directions, then the equation of the cone is:

(x²+y²+z²)cos²θ = (lx + my + nz)²

where,

• θ represents the Semi-Vertical Angle
• (l, m, n) are Direction Cosines of the Axis.

Examples of Right Circular Cone

Below are some real-world examples of right circular cones:

1. Ice cream cone: A classic example of a right circular cone is the ice cream cone. It has a pointed apex and a circular base, forming a conical shape that holds the ice cream.
2. Traffic cones: Traffic cones used on roads and construction sites are often shaped like right circular cones. They are typically orange and have a pointed top and a wide circular base for stability.
3. Party hat: Party hats often have a conical shape with a pointed top and a circular base. They are commonly worn during celebrations and festivities.
4. Volcano: The cone-shaped structure of a volcano is often modeled as a right circular cone. The lava erupts from the apex, creating a conical mound around the vent.
5. Speaker: Some speaker designs feature a conical shape for the speaker enclosure. This design helps in directing sound waves outward and enhances the acoustic performance.

Right Circular Cone Solved Questions

Question 1: Calculate the surface area when the radius and slant height of a right cone arecones 7 inches and 13 inches, respectively. (Use π = 22/7)

Solution:

Given 

Radius (r) = 7 inches

Slant Height (l) = 13 inches

We know that,

Surface Area of Right Cone (TSA) = πr(r + l) square units

TSA = 22/7 × 7 × (7 + 13) 

       = 22 × 20 

       = 440 sq. in

Hence, the Surface Area of the Right Cone is 440 sq. in.

Question 2: Find the curved surface area of a right cone if its radius is 7 units and its height is 24 units.

Solution:

Given,

Radius (r) = 7 units 

Height (h) = 24 units

We know that,

Curved Area of Right Cone (CSA) = πr√(h² + r²) square units

CSA = (22/7) × 7  × √(24² + 7²)

CSA = 22 × √(576 + 49)

CSA = 22 × 25 

CSA = 550‬ square units.

Hence, the Curved Surface Area of the Right Cone is 550 square units.

Question 3: Find the slant height of a right cone if its radius is 21 cm and its curved surface area is 660are sq. cm. (Use π = 22/7)

Solution:

Given,

Radius of Right Cone (r) = 14 cm

Curved Surface Area of Right Cone = 616 sq. cm

Let slant height of the right cone be l

We know that,

Curved Surface Area of Right Cone = πrl square units

660 = (22/7) × 21 × l

66 × l = 660

l = 660/66 = 10 cm

Hence, the slant height of Right Cone is 10 cm.

Question 4: Find the volume of a right cone if its radius is 21 units and its height is 8 units.

Solution:

Given,

Radius (r) = 21 units 

Height (h) = 8 units

We know that,

Volume of Right Cone = (1/3) × πr² × h

= (1/3) × 22/7 × (21)² × 8

= 3696 unit³

Thus, the Volume of Right Cone is 3696 unit³

Practice Questions on Right Circular Cone

1. If the radius of the base of a right circular cone is 6 cm and its height is 8 cm, what is the slant height of the cone?
2. A right circular cone has a slant height of 10 meters and a radius of 4 meters. What is the total surface area of the cone?
3. The volume of a right circular cone is 100π cubic centimeters, and its radius is 5 centimeters. What is the height of the cone?
4. A right circular cone has a height of 12 inches and a slant height of 15 inches. What is the radius of its base?
5. If the volume of a right circular cone is 200 cubic units and its height is 10 units, what is the radius of its base?

FAQs on Right Circular Cone

What is a right circular cone?

A cone in which the centre of the circular base is perpendicular to the vertex of the cone is called the right circular cone. We can obtain a right circular con by rotating a right triangle along its perpendicular or base.

What is the relationship between the slant height, height, and radius of a right circular cone?

The relation between slant height, height and radius of the right cone is,

l² = h² + r²

What is the volume of a right circular cone?

The volume of the right circular cone is the total space occupied by the cone. It is calculated using the formula,

V = 1/3 × πr² h

What is the surface area, or total surface area, of a right circular cone?

The total material required to build the cone is called the Total Surface Area of the cone. It is calculated by adding the CSA and the area of the circular base of the cone. Its formula is,

Total Surface Area of Right Circular Cone = π(r + l)r

What is the curved surface area of a right circular cone?

The area of the curved surface of the right circular cone is the Curved Surface area of the right circular cone. It is calculated using the formula,

Curved Surface Area of Right Circular Cone = πrl

What is the face of a right circular cone?

A cone has two faces:

• One flat face, which is the base, and
• One curved face, which is circular.

What is an example of a right circular cone?

An example of a right circular cone is an ice cream cone. It has a pointed apex, a circular base, and a curved lateral surface. Traffic cones, party hats, and volcano shapes are also examples of right circular cones.

What is the difference between the CSA of a cone and the TSA of a cone?

CSA (Curved Surface Area) of a cone refers to the area of the curved surface that wraps around the cone, excluding the base. The TSA (Total Surface Area) of a cone includes both the CSA and the area of the circular base. Mathematically, CSA = πrl, where r is the radius of the base and l is the slant height, while TSA = CSA + πr², where r is the radius of the base.

Is a cone a 3D shape?

Yes, a cone is a three-dimensional (3D) shape. It has a circular base, a curved lateral surface, and a pointed apex. Cones belong to the category of geometric solids in three-dimensional space.

What is the SA of a right circular cone?

For a right circular cone of height ‘h’, radius ‘r’ and slant height ‘l’, the SA is:

• Curved surface area of right circular cone = π r l
• Total surface area of a right circular cone = π(r + l)rIs a