Volume of Conical Cylinder

The volume of a conical cylinder is the space occupied by it. A conical cylinder is a three-dimensional shape known as an inverted frustum. It is formed when the vertex of a cone is cut by a plane parallel to the base of the shape and it is inverted. In this lesson, we will discuss the volume of a conical cylinder by using solved examples. Stay tuned for more!!!

The volume of a conical cylinder is the amount of space that is present inside it. A conical cylinder is a three-dimensional shape that is formed when the vertex of a cone is cut by a plane parallel to the base of the shape. On cutting the shape we obtain two parts, of which the part containing the base is known as the frustum of the shape. As it is a three-dimensional shape, thus, the volume of the conical cylinder also lies in a three-dimensional plane. The volume of the conical cylinder is expressed in the units m³, cm³, in³, or ft³etc.

The formula of volume of a conical cylinder can be calculated using the height of the conical cylinder and the base radius. We have two methods to obtain the formula for the volume of the conical cylinder. In both the methods, we consider the height of the cone as height H + h and base radius R. The conical cylinder is considered to have a height H with a small base radius "r" and the large base radius "R". Here, L and L + l are the slant heights of the conical cylinder and the cone respectively. Thus, the volume of the conical cylinder is:

• Volume of conical cylinder = πh/3 [ (R³ - r³) / r ]
• Volume of conical cylinder = πH/3 (R² + Rr + r²)

Consider the below figure, where, the height of the cone is (H + h), the base radius of the cone is R. When the cone is cut with a plane parallel to the base, the height of the conical cylinder is H, the smaller base radius is r, and larger base radius is R.

We know that volume of cone is, πR² (H + h) / 3.
The volume of the cone (with apex) that is cut is πr²h / 3.

We have, the volume of the conical cylinder, V = Volume of the cone - Volume of the cone that is cut
V = (πR² (H + h)/3) - (πr²h/3) ... (1)

As we know, the triangles OBC and PQC are similar,

(H + h) / h = R / r ... (2)
H + h = Rh / r ... (3)

Substituting (2) and (3) in (1),

V = πR² · (Rh / r) - πr²h / 3
⇒ V = πh/3 [ (R³ - r³) / r ] ... (4)

From (2) we get,

(H / h) + 1 = R / r
⇒ H / h = (R / r) - 1
⇒ H / h = (R - r) / r
⇒ h / H = r / (R - r)
⇒ h = (H r) / (R - r) ... (5)

Substituting (5) in the (4),

V = (π / 3) [ (H r) / (R - r) ] [ (R³ - r³) / r ]
By applying this formula to R³ - r³, using one of the algebraic formulas, a³ - b³ = (a - b) (a² + ab + b²).

V = (π / 3) [ (H r) / (R - r) ] [ (R - r) (R² + Rr + r²) / r ]
⇒ V = πH/3 (R² + Rr + r²)

We can determine the volume of a conical cylinder using the following steps:

• Step 1: Identify the height "H", of the conical cylinder.
• Step 2: Identify the value of larger base radius "R" and smaller base radius "r".
• Step 3: Use the formula volume of the conical cylinder, V = πH/3 (R² + Rr + r²) to determine the value of the volume of the conical cylinder.
• Step 4: Once the value of the volume of the conical cylinder is obtained, write the unit with it (in cubic units).

Example: Find the volume of a conical cylinder with a height of 8 units, a larger base radius of 10 units, and a smaller base radius of 5 units.

Solution: Given that H = 8 units, R = 10 units and r = 5 units

As we know the volume of the conical cylinder, V = πH/3 (R² + Rr + r²)
⇒ V = (π × 8)/3 × (10² + (10 × 5) + 5²)
⇒ V = 8.3775 × (100 + 50 + 25)
⇒ V = 8.3775 × (175) = 1,466.07 cubic units

Thus, the volume of the conical cylinder is 1,466.07 cubic units.

What is the Volume of a Conical Cylinder?

The volume of a conical cylinder is defined as the amount of space within the conical cylinder. It is expressed in cubic units where units can be, m³, cm³, in³ or ft³, etc. A conical cylinder is also known as a frustum.

What is the Formula for Volume of Conical Cylinder?

The formula for the volume of a conical cylinder is given as, V = πH/3 (R² + Rr + r²) where "V", "H", "R" and "r" are volume, height, larger base radius, and smaller base radius of the conical cylinder.

How to Find the Volume of Conical Cylinder?

We can use the following steps to determine the volume of the conical cylinder:

• Step 1: Identify the given height of the conical cylinder.
• Step 2: Identify the value of the larger base radius and the smaller base radius.
• Step 3: Use the formula of volume of the conical cylinder V = πH/3 (R² + Rr + r²) to find its volume.
• Step 4: The value so obtained is the volume of the conical cylinder and written the unit with it (in cubic units).

How to Find the Height of Conical Cylinder If the Volume of Conical Cylinder is Given?

We find the height of the conical cylinder if the volume of the conical cylinder is given by using the below steps:

• Step 1: Write the given dimensions of the conical cylinder.
• Step 2: Substitute the given values in the formula of volume of the conical cylinder, V = πH/3 (R² + Rr + r²) assuming the height of the conical cylinder as "H"
• Step 3: Solve for "H".
• Step 4: The value so obtained is the height of the conical cylinder.

What Happens to the Volume of Conical Cylinder If the Height of Conical Cylinder is Doubled?

If the height of the conical cylinder is doubled, the volume of the conical cylinder gets doubled as V = πH/3 (R² + Rr + r²) and we substitute "H" as "2H". Thus, the volume of the conical cylinder is V = π(2H)/3 (R² + Rr + r²) = 2πH/3 (R² + Rr + r²) which is 2 times the original volume of conical cylinder.