Right Circular Cone
A right circular cone is a type of cone whose axis falls perpendicular on the plane of the base. A cone is a 3D geometric figure that has a flat circular surface and a curved surface that meet at a point toward the top. The point formed at the end of the cone is called the apex or vertex, whereas the flat surface is called the base. Any triangle will form a cone when it is rotated, taking one of its two short sides as the axis of rotation.
What is a Right Circular Cone?
A right circular cone is a type of cone with an axis perpendicular to the plane of the base. A right circular cone is generated by a revolving right triangle about one of its legs. We can also observe this from the figure given below, the right-angled triangle when revolved results in the formation of a cone. The base of a right circular cone is in the shape of a circle.
Parts of a Right Circular Cone
The three elements that are present in the right circular cone are its radius, height, and slant height. The distance between the center of the circular base to any point on the circumference of the base of the right circular cone is defined as its radius, while the distance from the apex of the cone to the center of the circular base is called its height. The distance between the apex of the cone to any point on the circumference of the cone refers to as its slant height.
Right Circular Cone vs Oblique Cone
A cone can be classified into two types based on the alignment of the apex in comparison to the base: the right circular cone and the oblique cone. A right circular cone or regular cone's axis is perpendicular to its base, whereas the oblique cone appears to be tilted and its axis is not perpendicular to the base.
Another way to check if a cone is a right circular cone is to check its cross-section in a horizontal plane. A right circular cone will give a circular cross-section, whereas an oblique cone will give an oval cross-section.
Properties of a Right Circular Cone
There are certain properties of a right circular cone that distinguish it from other shapes. These properties are as listed below,
- It has a circular base. The axis is a line that joins the vertex to the center of the base.
- The slant height of the cone is measured from the vertex to the edge of the circular base. It is denoted by 'l' or 's'.
- The altitude or height of a right cone coincides with the axis of the cone and is represented by 'h'.
- If a right triangle is rotated with the perpendicular side as the axis of rotation, a right circular cone is constructed. The surface area generated by the hypotenuse of the triangle is the curved surface or the lateral surface area.
- Any horizontal section of the right circular cone parallel to the base produces the cross-section of a circle.
- A plane that passes through the vertex and any two points on the base of a right circular cone generates an isosceles triangle as given below:
Surface Areas of a Right Circular Cone
The surface area of a right circular cone is defined as the total region covered by the surface of the 3-D dimensional shape. It is expressed using square units, like cm2, m2, in2, ft2, etc. Let us cut and open up a right circular cone, to understand about the surface area. The curved surface forms a sector with radius 's', as shown below.
The surface area of a right circular cone can be of two types:
- Curved surface area or CSA
- Total surface area or TSA
Curved Surface Area of a Right Circular Cone
The curved surface area of a right circular cone is defined as the region occupied by the curved surface of the cone. We thus don't include the area of the base while referring to the curved surface area of a right circular cone. Curved surface area is also known as the lateral surface area.
Total Surface Area of a Right Circular Cone
The total surface of a right circular cone is defined as the region or area of the complete surface of the cone, including the base area. Let us understand the formulas to calculate the both CSA and TSA of a right circular cone in the next section.
Surface Area of a Right Circular Cone Formula
We discussed in the previous section that a right circular cone can have two surface areas, curved surface area or total surface area. We can calculate TSA and CSA for a right circular cone using different formulas.
Curved Surface Area Formula
The formula to calculate a right circular cone's CSA formula can be given as,
The curved surface area of a cone = Area of the sector with radius length equal to the slant height i.e., 's',
Curved surface area of a cone = πrs = πr√(r2 + h2)
where,
- r = Base radius
- h = Height of right circular cone
- s = Slant height of the right circular cone
Total Surface Area Formula
The formula to calculate a right circular cone's TSA formula can be given as,
The total surface area of a cone = Area of circular base + Curved surface area of a cone (sector's area)
Total surface area of a cone = πr2 + πrs
Total surface area of a cone = πr2 + πr√(r2 + h2)
where,
- r = Base radius
- h = Height of right circular cone
- s = Slant height of right circular cone
Note: Total surface area of a right circular sometimes can be referred to as only surface area. So, whenever we are asked to calculate the surface area, it means we have to find the total surface area of the given cone.
Volume of a Right Circular Cone
Volume of the right circular cone is defined as the total space occupied by the object in a 3-dimensional plane. The volume of a cone is expressed in cubic units, like in3, m3, cm3, etc. The volume of a right circular cone that has a circular base with radius 'r' and height 'h' will be equal to one-third of the product of the area of the base and its height. We can calculate the volume of the right circular cylinder, given the base radius and height, using the general formula.
Volume of a Right Circular Cone Formula
The volume of a right circular cone can be calculated using the base radius and height of the cone. We can observe from the image given below that the volume of a right circular cone is (1/3) times the volume of a right circular cylinder. The formula to find the volume of the right circular cone can be given as,
Volume of a Cone(V) = (1/3) × Area of Circular Base × Height of the Cone
V = (1/3) × πr2 × h
or,
V = (1/12) × πd2 × h
where,
- r = Base radius
- d = Diameter of base
- h = Height of right circular cone
Challenging Question:
Two children planned to dig a pit in the ground in the shape of a cone. They dug out 48 π cubic units of mud to form the pit. If the depth of the pit was 9 units, find the radius of the pit. Sam knows that the ratio of the volume of a cone and the volume of a cylinder with the same height and radius is 1:3. How will you help him prove that?
Important Notes:
- A cone has one circular face, no edge, and one apex (vertex).
- The axis of a right circular cone is perpendicular to the plane of the base.
- Curved surface area of a cone πrs = πr√(r2 + h2)
- Total surface area of a cone πr2 + πr√(r2 + h2)
- Volume of a cone (1/3) × πr2h
Related Topics
FAQs on Right Circular Cone
What is Meant By a Right Circular Cone?
A right circular cone is one whose axis is perpendicular to the plane of the base. A right circular cone is generated by revolving a right triangle about one of its legs.
What is the Volume of a Right Circular Cone?
Volume of a right circular cylinder is defined as the total space that is enclosed by a right circular cone in the three-dimensional plane. The volume of a Cone = (1/3) × πr2h, where, 'r' is the radius and 'h' is the height of the cone.
What is the Surface Area or Total Surface Area of a Right Circular Cone?
Total surface area of a cone is defined as the total area or region covered by the surface of a right circular cone, including the area of the circular base. It can be calculated using the formula, Total surface area of a cone = πr2 + πrs, where, 'r' is the radius, 's' is the slant height, and 'h' is the height of the cone.
What is the Curved Surface Area of a Right Circular Cone?
Curved surface area of a cone is defined as the area or region covered by the curved surface of a right circular cone. It can be calculated using the formula, Curved surface area of a cone = πrs = πr√(r2+h2), where 'r' is the radius, 's' is the slant height, and 'h' is the height of the cone.
How Can You Find the Radius of a Right Circular Cone Using Volume?
The radius of a cone refers to the radius of its circular base. The radius of a cone could be found using its volume and height. The formula for volume is given as, the volume of a cone = (1/3) × πr2h, where, 'r' is the radius and 'h' is the height of the cone. Substituting the known values, we can find the measure of radius.
How Many Vertices are There in a Right Circular Cone?
There is only one vertex in a right circular cone. Axis is the line joining this vertex to the center of the base of a right circular cone.
How Do You Find the Radius of a Right Circular Cone?
We can find the radius of a right circular cone using the formula in terms of its height and radius. The formula for this case can be given as, s = √(r2 + h2), where
- 'r' is the radius,
- 's' is the slant height, and,
- 'h' is the height of the cone.
How Do You Find the Side Length/Slant Height of a Right Cone?
The height of the cone, the slant height, and the radius of the base form a right triangle. So, we can use the Pythagorean theorem to find the slant height.
Is it Possible to Find the Right Circular Cone With the Same Height and Slant Height?
A right circular cone cannot have the same height as its slant height. If the slant height is considered as the hypotenuse of the right triangle, then we know that the length of the hypotenuse is greater than the lengths of the remaining two sides of the triangle.