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, cm3, 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:
- 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.
- 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.
- 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.
- 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.
- 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
- 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?
- 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?
- The volume of a right circular cone is 100π cubic centimeters, and its radius is 5 centimeters. What is the height of the cone?
- A right circular cone has a height of 12 inches and a slant height of 15 inches. What is the radius of its base?
- 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?
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