Volume of a Right Circular Cone: Formula, Derivation & Examples

Volume of a cone is the amount of space taken by a cone in a three-dimensional plane. A cone has a circular base, which means the base is made of radius and width. Then from the center of the base, you can go to the highest part of the cone which is measured as the cone’s height. In this article, we will learn about the formula, derivation, and examples of a circular cone. 

Key Terms: Volume, Right Circular Cone, Height, Radius, Width, Three dimensional Plane

Volume of a Right Circular Cone

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Volume of a right circular cone defines the area or capacity of a cone. A cone is a three-dimensional geometric shape with a rounded base that extends from a flat base to an apex or vertex. The cone is made up of a set of line segments, half-lines, or lines that connect a common point, apex, to all points on a plane base that does not contain an apex. 

Right Circular Cone

The cone can be seen as a set of inconsistent circular discs piled up on top of each other so that the radius of the nearest disk remains unchanged. Therefore, in the right circular cone, the vertex is above the center of the base vertically, while in the oblique cone, the vertex of the cone is not vertical above the base center. 

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Formula of a Right Circular Cone

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Generally, a cone is a pyramid with a circular cross. The right corner is a cone with its vertex above the base center. It is also called the right circular cone. You can easily find the volume of a cone if you have measurements of its height and radius. 

Thus, the formula of the volume of a right circular cone is given below:

Volume of a cone =\(\frac{1}{3}\) \(\pi\)r²h

Where, 

r = Base radius of a cone

h = Height of a cone

Volume of a cone formula

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Derivation of Volume of a Cone

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You can think of a cone as a circular triangle about one of its vertices. Now, think of a situation in which we need to calculate the amount of water that can be stored in a round flask. In other words, calculate the volume of this flask. The capacity of the conical flask is basically equal to the capacity of the cone involved. Therefore, the volume of a three-dimensional shape is equal to the amount of space occupied by that shape.

Volume of a Cone Derivation

Now, let’s suppose a cone has a circular base with radius ‘r’ and height ‘h’. Now, the volume of a cone will be equal to the \(\frac{1}{3}\)^th of the product of the area of the base and height. 

Thus, the volume of a cone V = \(\frac{1}{3}\) x Area of Circular Base x Height of the Cone

But, we know the formula of the area of a circle, the base of the cone has an area, which is equal to 

B = \(\pi\)r²

Thus, by substituting this value, we get,

V = \(\frac{1}{3}\) \(\pi\) r² h 

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Things to Remember

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• Volume of a cone is the amount of space taken by a cone in a three-dimensional plane. 
• A cone has a circular base, which means the base is made of radius and width.
• A cone is a solid 3-D shape with a rounded base. It has a curved surface.
• The volume of a cone = ? ????r²h
• The volume of the cone and the cylinder are related in the same way as the volume of the pyramid and the prism is related. When the cone and cylinder height are equal, the cylinder capacity is three times the cone capacity.

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