Volume of a Right Circular Cone: Definition, Terms, Derivation, Examples

The volume of a right circular cone is defined as the space occupied by the right circular cone. A right circular cone is a three-dimensional solid object which is having a circle at one end and a pointed end on the other and whose axis is perpendicular to the plane of the base. A right circular cone is obtained by a revolving right triangle about one of its legs. Read the complete article to learn about the definition and formulas to find out the volume of a right circular cone or the capacity of the right circular cone along with solved examples, practice questions and more.

Volume of a Right Circular Cone

Let the volume or the space occupied by the right circular cone be V. The volume of the right circular cone can be obtained by taking one-third of the product of the area of the circular base and its height or equal to the one- third the volume of a right circular cylinder of the same height and base radius. The formula for the volume of a right circular cone is V = (1/3) × πr²h; r is the radius of the base circle and h is the height of the cone. The common units of the right circular cone are cm³, m³, in³, or ft³, etc. The definition of various terms related to a right circular cone and derivation of the formula for volume are discussed in the sections below:

Definition of a Right Circular Cone

A cone is a three-dimensional shape with a circular base and narrows smoothly to a point above the base. This point is known as the apex.

Some of the most common examples of cones in our daily life are:

1. Ice cream cone
2. Funnel
3. Traffic cone
4. Party hat

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Difference Between a Cone and a Right Circular Cone

A right circular cone is a cone where the cone’s axis is the line meeting the vertex to the mid-point of the circular base. The centre of the circular base is joined with the apex of the cone, and it forms a right angle. A right circular cone is a cone in which the altitude or height is exactly perpendicular to the radius of the circle. In comparison, a cone is a \(3D\) figure with one curved surface and a circular base.

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Terms Related to a Right Circular Cone

Vertex: The point where the surface of the cone ends is called the vertex of the cone.

Height: The height of a cone is the length of its altitude.

Radius: Radius is the distance from the circle’s centre to its perimeter, also known as its circumference. The radius of a cone Is nothing but the radius of its flat circular base.

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Base: The area of the flat surface (bottom surface) of the cone.

Slant Height: The slant height of a cone is the distance along the curved surface, drawn from the edge at the top to a point on the circumference of the circular base.

Calculation of Slant Height: Using the Pythagoras theorem,

\({\text{hypotenus}}{{\text{e}}^2} = {\text{bas}}{{\text{e}}^2} + {\text{heigh}}{{\text{t}}^2}\)

Thus, \(l=\sqrt{h^{2}+r^{2}}\), where \(l, h\), and \(r\) are slant height, height, and radius of the base of the cone, respectively.

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Surface Area of a Cone: A right circular cone has two surfaces. One is the flat circular base, and the other one is the curved surface.

Surface Area and Volume of a Right Circular Cone

What is Surface Area?: The surface area of a solid object is a measure of the total area that the surface of the object occupies.

What is Volume?: Volume can be defined as the \(3\)-dimensional space enclosed by a boundary or occupied by an object.

Surface Area of a Right Circular Cone

Curved Surface Area of a Cone \(=\pi r l\), where \(r\) is the radius of the cone and \(l\) is the slant height.
Surface Area of a Cone \(=\pi r(r+l)\), where \(l\) is the slant height of the cone
Surface Area of a Cone \(=\pi r\left(r+\sqrt{\left(h^{2}+r^{2}\right)}\right)\), where \(r\) is the radius of the circular base \(h\) is the height of the cone.

Volume of a Right Circular Cone

Formula for Volume of a Solid Right Circular Cone: The formula for the volume of a right circular cone is given by \(\frac{1}{3} \pi r^{2} h\), where \(r\) is the radius of the cone and \(l\) is the slant height.

Volume of a Right Circular Cone Derivation: The volume of a cone involved is basically equal to the capacity of a conical flask. Thus, the amount of space occupied by that shape is equal to the volume of a three-dimensional shape. Let us perform an activity to derive the volume of a cone formula.

Let’s take a cylindrical container and a conical flask of the same base radius and same height. Add water to the conical flask such that it is filled to the brim. Let us add this water to the cylindrical container. We will notice it doesn’t fill up the container fully.

Repeat this experiment once more. We will still observe some vacant space in the container. Repeat this experiment once again; we will notice this time the cylindrical container is completely filled. Thus, the volume of a cone is equal to one-third of the volume of a cylinder having the same base radius and height.

Now let us derive its formula. Suppose a cone has a circular base with radius “\(r\)” and its height is “\(h\)”. The volume of the cone will be equal to one-third of the product of the area of the circular base and its height. Therefore,
\(V=\frac{1}{3} \times\) area of circular base \(\times\) height of the cone

By the formula of area of the circle, the base of the cone has an area (say \(C\)) equals to;
\(C=\pi r^{2}\)

Hence, substituting this value, we get;
\(V=\frac{1}{3} \times \pi r^{2} \times h\)

where \(V\) is the volume of the cone, \(r\) is the radius, and \(h\) is the height.

Solved Examples – Volume of a Right Circular Cone

Q.1. Find the volume of a cone whose radius is \(3 \mathrm{~cm}\) and height is \(7 \mathrm{~cm}\) (Use \(\pi  = \frac{{22}}{7}\))
Ans: As we know, the volume of the cone is \(\frac{1}{3} \pi r^{2} h\).
Given that: \(r=3 \mathrm{~cm}, h=7 \mathrm{~cm}\) and \(\pi=\frac{22}{7}\)
Thus, the volume of cone, \(V=\frac{1}{3} \pi r^{2} h\)
\(\Rightarrow V=\frac{1}{3} \times \frac{22}{7} \times 3^{2} \times(7)=22 \times 3=66 \mathrm{~cm}^{3}\)
\(\therefore\) The volume of cone is \(66 \mathrm{~cm}^{3}\).

Q.2. A Metallic right circular cone of volume \(264 \mathrm{~cm}^{3}\) has the height \(7 \mathrm{~cm}\). Find the radius of the cone (Use \(\pi  = \frac{{22}}{7}\)).
Ans: As we know, the volume of the cone is \(\frac{1}{3} \pi r^{2} h\).
Given that: \(h=7 \mathrm{~cm}\) and \(\pi=\frac{22}{7}, V=264 \mathrm{~cm}^{3}\)
Thus, the volume of cone, \(V=\frac{1}{3} \pi r^{2} h\) \(\Rightarrow 264=\frac{1}{3} \times \frac{22}{7} \times r^{2} \times(7)\)
\(\Rightarrow r^{2}=264 \times \frac{3}{22}\)
\(\Rightarrow r=\sqrt{36}=6 \mathrm{~cm}\)
\(\therefore\) The radius of the cone is \(6 \mathrm{~cm}\).

Q.3. Meena is filling a conical box with gems. She knows the capacity of each box is \(24 \pi \,\mathrm{m}^{3}\). Help her to find the height of the conical box of radius \(3 \mathrm{~m}\).
Ans: The given dimensions are the radius of the conical bag \(=3 \mathrm{~m}\), volume of cone \(=24 \pi\, \text {m}^{3}\) and let the height of the cone \(x\,{\rm{m}}.\)
Substituting the given values in the volume of a Cone formula
Volume of Cone \(=\frac{1}{3} \pi r^{2} h=\left(\frac{1}{3}\right) \times \pi \times 3^{2} \times x=24 \pi\,\text {m}^{3} \Rightarrow 3 x=24 \Rightarrow x=8 \mathrm{~m}\)
\(\therefore\) The height of the conical box is \(8 \mathrm{~m}\)

Q.4. If the height of a cone is \(15 \mathrm{~cm}\) and its volume is \(770 \mathrm{~cm}^{3}\), find the radius of its base.
Ans: Given, \(h=15 \mathrm{~cm}\) and \(V=770 \mathrm{~cm}^{3}\)
Volume of cone \(=\frac{1}{3} \pi r^{2} h\)
\( = \;\frac{1}{3}\pi {r^2}h\,\,\, \Rightarrow \,\,\,\;770\; = \;\frac{1}{3} \times 3.14\,\; \times \,{r^2} \times 15\)
\( \Rightarrow \;\,\,770\; = \;3.14\; \times \,\;{r^2}\; \times \;5\;\)
\(\Rightarrow \;\,\,770\; = \;15.7\; \times \;{r^2}\)
\( \Rightarrow \;\,\,{r^2}\;\; = \frac{{770}}{{15.7}} = \;49\)
\( \Rightarrow \;\,\,{r^2}\; = \;49\)
\(\therefore \;\,\,\,r\; = \;7\;{\rm{cm}}\)

Q.5. Calculate the total surface area of a cone whose radius is \(8 \mathrm{~cm}\) and height is \(12 \mathrm{~cm}\).
Ans: We know that the total surface area is given as \(\pi r(r+l)\)
Also, \(l=\sqrt{r^{2}+h^{2}}\)
Also, \(l=\sqrt{8^{2}+12^{2}}=\sqrt{208}=14.42\)
So, the total surface area of the cone \(=\pi(8)(8+14.42)\)
\(=\pi(8)(22.42)\)
\(=179.36 \pi \,\mathrm{cm}^{2}\)
Thus, the whole or total surface area of the cone \(=179.36 \pi\, \mathrm{cm}^{2}\)

Summary

In this article, we have learnt the definition of cone and the right circular cone, and we discussed daily-life examples of the right circular cone. We also studied terms related to the right circular cone-like vertex or apex, base, radius, height and slant height, etc., and we learnt the difference between a cone and a right circular cone.

Here, we studied the definitions of the surface area, the volume of the right circular cone, and its formulas. We derived the formula for the volume of the right circular cone and solved examples related to the volume of a right circular cone.

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Frequently Asked Questions (FAQ) – Volume of a Right Circular Cone

The most commonly asked doubts on Volume of a Right Circular Cone are answered here:

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