## Volume of a ConeIn this section, we will learn what is the cone, types of cone, the formula of ## ConeThe cone is a three-dimensional geometrical shape that has a circular base (surface) and a vertex. The vertex is connected with the base through the two slanted line segments. These two-line segments connected at a common point called the vertex. **Vertex:**It is the pointed part of the cone that is just above the base.**Slant Height:**It is the height of slanted line-segments that meet at the vertex. It is denoted by l.**Height:**The perpendicular distance from base to the vertex is called the height of a cone. It is also known as perpendicular height. It is denoted by h.**Base:**It is a circular surface of a cone.**Radius:**The distance between the center and the circumference is called the radius. It is denoted by r.
## Types of ConeThere are two types of cones: - Right Cone or Cone
- Oblique Cone
## The volume of a ConeIn mathematics, the area enclosed by an object is called the ## Note: The radius and height must be in the same unit. Convert the units before calculation if they are different.## Unit of the VolumeThe unit of the volume is a ## The Formula of the Volume of a ConeThe volume (V) of a cone with radius (r) is one-third the area of the base time height. In other words, the volume of a cone is ## Note: The above formula is also used to find the volume of an oblique cone.The volume of the cylinder will be three times the volume of the cone if the height of the cylinder and cone are equal.
## Derivation of the FormulaIn the following figure, we have a cylindrical shape and a conical shape of the same height and radius. Put the conical shape is inside the cylindrical shape. Now pour the water in the cylindrical shape. We see that it does not fill the cylinder up to the capacity. After repeating this process two to three times, we see that the same experiment fills the cylindrical shape up to the capacity. Therefore, the volume of a cone is equal to the one-third volume of a cylinder. Consider the above figure, the radius of the circular base is r, and the height is h. We know that the volume of a cylinder is the product of the area of the base and its height. The Volume of a Cylinder (V)=Area of the base×height of cylinder
We know that the volume of a cone is equal to the one-third volume of a cylinder. Hence, The Volume of a Cone (V)=×Volume of Cylinder Where:
Let's see how to find the volume of a cone.
Given, height (h)= 15 cm radius (r) = 4 cm We know the formula of volume of a cone: Putting the values in the above formula, we get:
Given, height (h)= 19 m diameter = 10 m We know that the radius is half the diameter. Hence, Putting the values in the above formula, we get:
Given, slant height (l) = 30 cm radius (r) = 12 cm π=3.14 We know the formula of volume when slant height and radius are given. Putting the values in the above formula, we get:
Given, the volume of a cylinder (V) = 223 m The volume of a cone (V) =? We know that the volume of the cone is one-third of the volume of the cylinder. Therefore: V=×223
Given, height (h) = 12 cm radius (r) = 4 cm We know that,
Putting the values in the above formula, we get: V=3.14×4 We also know that, The Volume of a Cone (V)=×Volume of Cylinder Putting the value in the above formula, we get: The Volume of a Cone (V)=×602.88
To verify the answer, multiply the volume of the cone by 3, we get the volume of the cylinder. Hence,
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