Volume Of A Slanted Cylinder

Volume of Conical Cylinder

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

1.	What is Volume of Conical Cylinder?
2.	Formula of the Volume of a Conical Cylinder
three.	Derivation of Volume of Conical Cylinder
4.	How to Find the Book of a Conical Cylinder?
5.	FAQs on Book of Conical Cylinder

What is Volume of Conical Cylinder?

The book of a conical cylinder is the amount of space that is present within 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 2 parts, of which the part containing the base is known as the frustum of the shape. As it is a iii-dimensional shape, thus, the volume of the conical cylinder also lies in a 3-dimensional aeroplane. The volume of the conical cylinder is expressed in the units m^three, cm³, in^three, or ftⁱⁱⁱetc.

Volume of Conical Cylinder - Formation

Formula of Book of a Conical Cylinder

The formula of volume of a conical cylinder tin be calculated using the top of the conical cylinder and the base radius. We have two methods to obtain the formula for the book of the conical cylinder. In both the methods, we consider the height of the cone equally elevation H + h and base radius R. The conical cylinder is considered to accept a meridian H with a minor base radius "r" and the big base of operations radius "R". Hither, L and 50 + 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 [ (R^three - r^three) / r ]
Book of conical cylinder = πH/3 (R² + Rr + r²)

Derivation of Volume of a Conical Cylinder

Consider the beneath effigy, where, the superlative of the cone is (H + h), the base of operations radius of the cone is R. When the cone is cutting with a aeroplane parallel to the base, the elevation of the conical cylinder is H, the smaller base radius is r, and larger base of operations radius is R.

Volume of Conical Cylinder - Formula

We know that book of cone is, πR² (H + h) / iii.
The book of the cone (with noon) that is cut is πr²h / three.

We take, the volume of the conical cylinder, 5 = Volume of the cone - Volume of the cone that is cut
V = (πR² (H + h)/3) - (πr²h/3) ... (1)

Equally 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 (one),

V = πR² · (Rh / r) - πrⁱⁱh / three
⇒ 5 = πh/iii [ (Rⁱⁱⁱ - r³) / r ] ... (4)

From (2) we get,

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

Substituting (5) in the (iv),

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

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

How to Notice the Book of a Conical Cylinder?

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

Pace 1: Identify the height "H", of the conical cylinder.
Stride 2: Identify the value of larger base radius "R" and smaller base radius "r".
Step 3: Apply the formula volume of the conical cylinder, V = πH/three (R² + Rr + rⁱⁱ) to determine the value of the volume of the conical cylinder.
Step 4: In one case the value of the volume of the conical cylinder is obtained, write the unit with it (in cubic units).

Instance: Detect the volume of a conical cylinder with a superlative of 8 units, a larger base radius of 10 units, and a smaller base of operations 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 (Rⁱⁱ + Rr + r²)
⇒ V = (π × 8)/3 × (10ⁱⁱ + (10 × 5) + 5ⁱⁱ)
⇒ 5 = viii.3775 × (100 + 50 + 25)
⇒ V = eight.3775 × (175) = 1,466.07 cubic units

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

Examples on Volume of Conical Cylinder

Example one: Find the meridian of the conical cylinder with a larger base radius of 7 inches and a smaller base radius of 4 inches if the volume of the conical cylinder 62π.

Solution: Given that R = 7 inches, r = 4 inches and V = 62π. Let the tiptop of the cylinder is "H".

Volume of the conical cylinder, V = πH/three (R² + Rr + r²)
⇒ 62π = πH/3 × (7^two + 28 + 4²)
⇒ 62 = H/3 × (49 + 28 + 16)
⇒ 62 × 3 = H × 93
⇒ H = (62 × iii)/93 = 2 inches

Therefore, the peak of the conical cylinder is 2 inches.
Instance ii: Detect the book of the conical cylinder whose height is 9 anxiety, the larger base radius is 10 anxiety and the smaller base radius is half-dozen anxiety.

Solution: Given that, H = nine anxiety, R = x anxiety and r = half dozen anxiety

Nosotros know that 5 = πH/3 (R² + Rr + r^two)
⇒ 5 = 9π/3 × (100 + 60 + 36)
⇒ V = 3π × 196
⇒ Five = 588π = ane,847.25 cubic anxiety

Therefore, the volume of the conical cylinder is 1,847.25 cubic feet.

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Practice Questions on the Volume of a Conical Cylinder

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FAQs on the Book of a Conical Cylinder

What is the Book 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, cmⁱⁱⁱ, inⁱⁱⁱ or ftⁱⁱⁱ, etc. A conical cylinder is also known as a frustum.

What is the Formula for Volume of Conical Cylinder?

The formula for the book of a conical cylinder is given as, V = πH/3 (R^two + Rr + r²) where "5", "H", "R" and "r" are volume, height, larger base radius, and smaller base of operations radius of the conical cylinder.

How to Find the Volume of Conical Cylinder?

We can use the post-obit steps to determine the book of the conical cylinder:

Footstep one: Identify the given top of the conical cylinder.
Step 2: Place the value of the larger base radius and the smaller base radius.
Step 3: Employ the formula of volume of the conical cylinder V = πH/3 (R² + Rr + rⁱⁱ) to find its book.
Step 4: The value and so obtained is the volume of the conical cylinder and written the unit of measurement with it (in cubic units).

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

We discover the peak of the conical cylinder if the book of the conical cylinder is given by using the beneath 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, 5 = πH/3 (R² + Rr + r²) bold the height of the conical cylinder equally "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 pinnacle of the conical cylinder is doubled, the volume of the conical cylinder gets doubled as Five = πH/iii (Rⁱⁱ + Rr + rⁱⁱ) and we substitute "H" as "2H". Thus, the volume of the conical cylinder is V = π(2H)/3 (Rⁱⁱ + Rr + r²) = 2πH/3 (R² + Rr + r^two) which is ii times the original volume of conical cylinder.