A right circular cone is a three-dimensional solid that has a circular base and a single vertex (apex), in which the line joining the center of the base to the apex is perpendicular to the plane of the base.
Parts of a Right Circular Cone
1. Radius of the Base (r): The radius of the base is the distance from the center of the circular base to any point on its circumference. It is denoted by r.
2. Height of the Right Circular Cone (h): The height of a right circular cone is the perpendicular distance from the center of the base to the vertex (apex) of the cone. It is denoted by h.
3. Slant Height of the Right Circular Cone (l): The slant height is the distance from the vertex to a point on the circumference of the base, measured along the curved surface of the cone. It is denoted by l.
Right Cone vs 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.
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 is a three-dimensional shape with a circular base and a vertex, having the following properties:
- A right circular cone is a three-dimensional solid with one circular base and one vertex (apex).
- The axis of the cone is the straight line joining the vertex to the center of the circular base.
- The axis (altitude) is perpendicular to the base and represents the height (h) of the cone.
- The cone can be formed by rotating a right-angled triangle about one of its perpendicular sides.
- The curved surface of the cone is generated by the hypotenuse of the rotating right-angled triangle.
- The slant height (l) is the distance from the vertex to any point on the circumference of the base, measured along the curved surface.
- Any section of the cone parallel to the base is a circle.
- A section passing through the vertex and the base forms an isosceles triangle.
Surface Area of a Right Cone
The surface area of a right circular cone is the total area covered by its surface and is measured in square units such as cm², m², etc.
- 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:
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
Area of circular base = πr²
Curved surface area (CSA) = πrlSo,
TSA = πr² + πrl
TSA = πr (r + l)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 volume of a right circular cone represents the amount of space occupied by the cone.
It is measured in cubic units such as cm³, m³, etc.
The volume of a right circular cone is one-third of the volume of a cylinder having the same base radius and height.
Volume of right cone = (1/3) × Volume of cylinder
For calculating the formula for volume of the right cone, we define the volume as:
V = (1/3) × Area of circular base × Height of the cone
We know that,
Area of circular base = πr²
Height of the cone = hSo,
V = (1/3) πr²hwhere,
r = radius of the base
h = height of the 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
Equation of Right Circular Cone
The equation of the right circular cone with vertex origin is:
(x2+y2+z2)cos2θ = (lx + my + nz)2
Where θ is the semi-vertical angle and (l, m, n) are direction cosines of the axis.
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
- 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?