Cone
A cone is a three-dimensional shape that has a circular base and it narrows down to a sharp point called a vertex. One of the easiest real-life examples that could be given is a birthday cap in the shape of a cone. With regards to a cone, we have two types of areas. One is the total surface area and the other is a curved surface area. The total surface area of a cone is defined as the area covered by its base and the curved part of the cone, whereas the curved surface area is defined as the area of the curved surfaces of the cone only.
| 1. | Cone Definition |
| 2. | Properties of Cone |
| 3. | Cone Formula |
| 4. | Types of Cone |
| 5. | FAQs on Cone |
Cone Definition
A cone is a three-dimensional solid geometric shape having a circular base and a pointed edge at the top called the apex. A cone has one face and a vertex. There are no edges for a cone.
The three elements of the cone 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 circumference of the base. Height 'h' of the cone is defined as the distance between the apex of the cone to the center of the circular base. The slant height 'l' is defined as the distance between the apex of the cone to any point on the circumference of the cone. The figure given below shows how the radius, height, and slant height of a cone will look like. Some of the real-life examples of a cone include a birthday cap, a tent, and a road divider.
Properties of Cone
A cone is a shape that has a curved surface and a circular base. The following properties of a cone help us identify it easily. They are as follows.
- A base of a cone is circular.
- There is one face, one vertex, and no edges for a cone.
- The slant height of a cone is the length of the line segment joining the apex of the cone to any point on the circumference of the base of the cone.
- A cone that has its apex right above the circular base at a perpendicular distance is called a right circular cone.
- A cone that does not have its apex directly above the circular base is an oblique cone.
Cone Formula
There are three important formulas related to a cone. They are the slant height of a cone, the volume of a cone, and its surface area. The slant height of a cone is obtained by finding the sum of the squares of radius and the height of the cylinder which is given by the formula given below. slant height2 = radius2 + height2. If the slant height of the cone is 'l' and the height is 'h' and the radius is 'r', then l2 = r2 + h2. The formula for the slant height of the cone is 'l' = \(\sqrt{r^2 + h^2}\)
Curved Surface Area of Cone
The curved surface area of a cone is the area enclosed by the curved part of the cone. For a cone of radius 'r', height 'h', and slant height 'l', the curved surface area is as follows:
Curved Surface Area = πrl square units.
Total Surface Area of Cone
Total surface area is the sum of the area of the circular base and the area of the curved part of the cone. In other words, it is the sum of the curved surface area of the cone and the area of the circular base, which can be written mathematically as:
Total Surface Area (TSA) = Area of the base (Circle) + Curved Surface Area of the Cone(CSA).
TSA = (πr2 + πrl) square units.
Total surface area is sometimes referred to as only surface area. So, whenever we are asked to calculate the surface area of the cone, it means we have to find the total surface area.
Volume of a Cone
The volume of a cone is the space occupied by the cone. The formula to find the volume of a cone, whose radius is 'r' and height is 'h' is given as, Volume = (1/3) πr2h cubic units. Let A = Area of base of the cone and h = height of the cone. Therefore, the volume of cone= (1/3) × A × h. Since the base of the cone is circular, we substitute the area to be πr2. Volume of cone = (1/3) × π × r2 × h cubic units. Also, the volume of a cone is one-third of the volume of a cylinder.
Volume of cone = (1/3) × volume of a cylinder.
Types of Cone
Broadly there are two types of Cones. One is the right circular cone and the other is an oblique cone. The table below lists out a few differences in how these two types of cones differ.
| Right Circular Cone | Oblique Cone |
|---|---|
| A right circular cone has its vertex opposite to the circular base. | An oblique cone does not have its vertex directly opposite to the circular base. |
| The line representing the height of the cone passes through the center of the base circle and is perpendicular to the radius. | The line representing the height of the cone does not pass through the center of the base circle. |
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FAQs on Cone
What is a Cone?
A cone is a three-dimensional figure which has a circular base and a curved surface. The pointed tip at the top of the cone is called 'Apex'. The cone has one face (which is circular) with no edges and one vertex, which is the apex of the cone.
How Many Faces, Edges, and Vertices Does a Cone Have?
A cone has one face, with no edges and one vertex. The cone has a circular base and a curved surface. Since a cone has only one vertex it does not have any edge. Also in a cone, there is only one flat surface that forms the base.
What is the Slant Height of a Cone?
The distance from the apex or top of the cone to a point on the circumference of the base is called slant height. The slant height is obtained by the square root of the sum of the squares of the radius and the height of the cone.
What is the Surface Area of the Cone?
The surface area of a cone can be obtained by adding the area of its base and its curved surface. The base of the cone is circular. The formula for finding the surface area of a cone is (πr2 + πrl) square units. Here, 'r' is the radius of the cone and 'l' is the slant height of the cone. Here, πr2 is the area of its base, and its curved surface area is πrl.
What is the Volume of a Cone?
The volume of a cone is the amount of space occupied by a cone. A cone with radius 'r' and height 'h' has a volume of (1/3)πr2h.
What Is the Base Area of a Cone?
The base of the cone is the quantity that shows the area covered by the circular base of the cone. The formula for the base area of a cone is A = πr2, where r is the radius of the base of the cone.
What are the Two Types of Cone?
The two types of cones are a right circular cone and an oblique cone. A right circular cone has the axis line that passes through the center of the circular base, whereas, in an oblique cone the axis line does not pass through the center of the circular base.
What is the Use of Cone Calculator?
In geometry, a cone is defined as a three-dimensional solid geometric figure having a circular base at one end and a pointed edge at the other end. Try now Cuemath's Cone Calculator', an online tool that helps you to calculate the parameters like surface area, volume, and slant height of the cone within a few seconds.