Volume of a Partial Cone

To understand the concept, the volume of a partial cone, we must first understand the shape of a cone and how a partial cone is formed from it. A cone is a 3-D geometric shape. It has an apex and a circular base. The apex is also called the vertex of the cone. A cone is formed by a set of line segments or lines connecting a common point, that is the apex, to all the points on a base that is in a plane that does not contain the apex. In general, we use the 'right circular' cone, which means that the axis of the cone passes through the center of the base at right angles to its plane. When a right circular cone is sliced along its cross-section, a smaller right circular cone and a partial cone is obtained.
Now that we know, how a partial cone gets formed, let's use this to find the volume of a partial cone and what is its formula?

A right circular cone has an apex and a circular base such that the axis of the cone passes through the apex and the center of the base at right angles. A partial cone has two circular faces and the axis of a partial cone passes through the centers at right angles of both the circular faces. A partial cone does not have an apex point. A partial cone is also known as a frustum.

The volume of a cone is defined as the number of cubic units occupied by the cone. Therefore, the unit of 'volume' is "cubic units". For example, it can be expressed as m³, cm³, in³, etc depending upon the given units. Let's see how do we find the volume of a cone.

If a cylinder inscribes a cone such that the base and the height of the cone and the cylinder is the same, then the volume of the cylinder and the volume of the cone are compared as given below:

Volume of a cone = 1/3 × Volume of a cylinder

or, Volume of a cone = 1/3 × Base area × Height

Since the base and the radius of both the cone and the cylinder are the same. Also since the base of the cone is a circle, therefore, base radius = πr²

or, Volume of a cone = 1/3 × πr² × h

The volume of a partial cone is derived from the volume of a whole cone based on the idea that how a partial cone is formed from a whole cone. Since a whole cone is sliced to give a partial cone and a small cone.

Therefore, Volume of a partial cone = volume of the whole cone - Volume of the small cone

or, Volume of the partial cone = 1/3 × Base radius of the whole cone × height of the whole cone - 1/3 × Base radius of the small cone × height of the small cone

Derivation of the Formula of the Volume of a Partial Cone

Let's now derive the formula of the volume of a partial cone based on the result that we have got.

Now, for the whole cone:

Height = 'H' units

Base radius = 'R' units

For the small cone:

Height = 'H - h' units

Base radius = 'r' units

Volume of a partial cone = volume of the whole cone - Volume of the small cone

or, Volume of the partial cone = 1/3 × Base radius of the whole cone × Height of the whole cone - 1/3 × Base radius of the small cone × Height of the small cone

= 1/3 × πR² × H - 1/3 × πr² × (H - h)

= 1/3 × π × (R² × H - r² × (H - h))

= 1/3 × π(R²H - r²H + r²h)

= 1/3 × π(H(R² - r²) + r²h)

Since the ratio of the height of the whole cone and the height of the small cone is H:(H - h) = R:r, therefore, the ratio of the height of the whole cone and the height of the partial cone is H:h = R:(R - r)

H/h = R/(R - r)

Multiplying h on both sides,

H = Rh/(R - r)

Substituting this value of H in the above equation, we get:

Volume of the partial cone, V = 1/3 × π(H(R² - r²) + r²h)
⇒ V = 1/3 × π[(Rh/(R - r)) × (R² - r²) + r²h]
⇒ V = 1/3 × π[(Rh/(R - r)) × (R - r)(R + r) + r²h]
⇒ V = 1/3 × π(Rh(R + r) + r²h)
⇒ V = 1/3 × π(R²h + Rrh + r²h)
⇒ V = 1/3 × π × h × (R² + Rr + r²)
⇒ V = 1/3 × πh(R² + Rr + r²)

Let us now have a look at a few solved examples on volume of a partial cone for better understanding.

What Is the Formula of the Volume of the Partial Cone?

The formula to calculate the volume of a partial cone is given as, Volume of a partial cone, V = 1/3 × πh(R² + Rr + r²), where, 'r' and 'R' are the base radii, such that R > r, and 'h' is the height of frustum.

What Are the Units Used When You Find the Volume of a Partial Cone?

The unit of 'volume' is "cubic units". For example, it can be expressed as m³, cm³, in³, etc depending upon the given units.

How Can We Find the Volume of a Partial Cone?

The volume of a partial cone can be calculated as: Volume of a partial cone = Volume of the whole cone - Volume of the small cone

What is the Formula of the Volume of the Cone?

The formula to calculate the volume of a cone is given as, Volume of a cone = 1/3 × πr² × h, where, r is the radius and h is the height of the cone.

How Do We Differentiate Between a Cone and a Partial Cone?

A cone has a circular base and an apex whereas a partial cone has two end circular faces.

How a Partial Cone Is Formed?

A partial cone is formed when a whole cone is sliced horizontally to give a small cone and a partial cone.