Volume of a cylinder is a fundamental concept in geometry and plays a crucial role in various real-life applications. It is a measure which signifies the amount of material the cylinder can carry. It is also defined as the space occupied by the Cylinder. The formula for the volume of a cylinder is πr2h where r is the radius of the base and h is the height of the cylinder.
In this article, we will explain the formula, highlight its applications, and provide examples to help you master how to calculate the volume of a cylinder efficiently for real-world use.
What is a Cylinder?
Cylinder is defined as a 3-D figure in which the base are circle and both surfaces are connected by a curved surface.
Cylinder is a three-dimensional geometric shape with two parallel and congruent circular bases connected by a curved surface. It has a fixed height, which is the perpendicular distance between the two circular bases, and a radius, which measures the distance from the center to the edge of each base.
For example, Gas cylinders and Rolling Pin in our houses resemble cylindrical shapes.
Volume of Cylinder
Volume of a cylinder is a crucial mathematical concept used to determine the space a cylindrical object occupies. It can also define the total capacity of any cylinder, i.e. the total amount of liquid any cylinder can hold. It is generally measured in liters, volume can also be measured in m3, cm3, etc. Volume of Cylinder is calculated by multiplying area of circular base (πr2) of cylinder by its height (h).
Volume of Cylinder
Volume of Cylinder is equal to πr2h. Where, r is the radius of radius of the base and h is the height of the cylinder.
V Cylinder = πr2h
Volume of Solid Cylinder
A Solid cylinder is a three-dimensional geometric cylinder with the space inside the cylinder completely filled. Unlike a hollow cylinder, where the interior has empty space, a solid cylinder has no empty space inside, meaning it is completely made up of material throughout its volume.
Volume of a Solid cylinder is also equal to the amount of space taken by it in a three-dimensional space. In other words, it determines the space or region enclosed by the cylinder is its volume, and the unit of volume is cubic unit i.e., the number of unit cubes (cubes of unit length) that may fit into an object.
Volume of Solid Cylinder : V Solid Cylinder = πr2h
Volume of a Right Circular Cylinder
In a Right Circular Cylinder angle between the plane of the base or top and the curved surface is a right angle. For a right circular cylinder, the base is a circle with radius r, thus its area is πr2, and the height of the cylinder is h then the volume of the cylinder is:
Volume of Right Circular Cylinder : V Right Circular Cylinder = πr2h
Where,
- r = radius of the base
- h= height of the cylinder
Volume of Right Circular Cylinder
Volume of a Hollow Cylinder
A hollow cylinder is a three-dimensional geometric shape that consists of two concentric cylindrical surfaces, where the space between the outer and inner surfaces is filled with material, but the interior part is hollow. thus, its bases form a ring with two radii, inner radius and an outer radius. In simple words a hollow cylinder is a cylinder, which is empty from the inside and has some difference between the internal and external radius. Suppose a hollow cylinder is taken with its inner radius as r and outer radius as R and the height of the cylinder is h then volume of hollow cylinder is given as:
Volume of Solid Cylinder : V Hollow Cylinder = π( R2 – r2)h
Where,
- R = Outer Radius
- r = inner radius
- h= height of the cylinder
Volume of a Hollow Cylinder
Volume of an Oblique Cylinder
An oblique cylinder is a type of cylinder where the sides are slanted, meaning the axis that connects the centers of the two circular bases is not perpendicular to the bases. In contrast to a right cylinder, where the axis is at a 90-degree angle to the bases, an oblique cylinder appears tilted.
In an Oblique Cylinder angle between the plane of the base or top and the curved surface is not a right angle.
But volume of oblique cylinder is given by the same formula as a product of the area of the base and the height of the cylinder.
Volume of Oblique Cylinder : V Oblique Cylinder = πr2h
Where,
- r = radius of the base
- h= height of the cylinder
Volume of Oblique Cylinder
Volume of an Elliptic Cylinder
An Elliptic cylinder is a three-dimensional geometric shape similar to a regular cylinder but with elliptical (oval-shaped) bases instead of circular ones. In an elliptic cylinder, the cross-section along its length is an ellipse, and the sides are perpendicular to the bases.
Volume of Elliptic Cylinder : V Elliptic Cylinder = πabh
Where,
- a = semi major axis of ellipse
- b = semi minor axis of ellipse
- h = height of the cylinder
Volume of Elliptic Cylinder
Volume of Cylinder in Liters
Generally, the volume of a cylinder is calculated in cubic meters or cubic centimeters but we can change them in litres by using the conversion factor discussed below i.e.,
1 cm3 = 1 ml
1000 cm3 = 1 litre
1 m3 = 1000000 cm3 = 1000 litres
Example: If a cylinder has a volume of 32 m3 it can be written as 32×1000 litres = 32000 litres.
Some formulas for cylinder other than volume formulas are,
Surface Area of a Cylinder
|
|
CSA ( Curved Surface Area)
|
Area of the surface except the base and top
|
CSA of Cylinder = 2πrh
|
|
TSA ( Total Surface Area)
|
Area of the complete surface of the cylinder
|
TSA of Cylinder = 2πr(r+h)
|
Volume of a Cylinder
|
|
Volume of a Cylinder
|
πr2h ( r= radius of base, h= height of cylinder)
|
|
Volume of Hollow Cylinder
|
π( R2 – r2)h ( r= inner radius, R= outer radius, h= height of cylinder)
|
|
Volume of Oblique Cylinder
|
πr2h ( r= radius of base, h= height of cylinder)
|
|
Volume of Elliptic Cylinder
|
πa b h ( a = semi major axis of ellipse, b = semi minor axis of ellipse, h= height of cylinder)
|
Read More
Examples Questions on Volume of Cylinder
Example 1: Calculate the volume of a cylinder of radius 3 m and a height of 4 m. (take π = 3.14)
Solution:
We have, r = 3 and h = 4
Using the formula we have,
V = πr2h
⇒ V = 3.14 × (3)2 × 4
⇒ V = 113.04 m3
Example 2: Calculate the volume of a cylinder of radius 4 m and a height of 7 m.
Solution:
We have, r = 4 and h = 7
Using the formula we have,
V = πr2h
⇒ V = 3.14 × (4)2 × 7
⇒ V = 351.68 m3
Example 3: Calculate the radius of a cylinder if its volume is 300 m3 and height is 7 m.
Solution:
We have, V = 300 and h = 7
Using the formula we have,
V = πr2h
⇒ r2 = V/πh
⇒ r2 = 300/(3.14 × 7)
⇒ r = 3.68 m
Example 4: Calculate the radius of a cylinder if its volume is 450 m3 and its height is 9 m.
Solution:
We have, V = 450 and h = 9
Using the formula we have,
V = πr2h
⇒ r2 = V/πh
⇒ r2 = 450/(3.14 × 9)
⇒ r = 12.52 m
Example 5: Calculate the height of a cylinder if its volume is 570 m3 and its radius is 4 m.
Solution:
We have, V = 570 and r = 4
Using the formula we have,
V = πr2h
⇒ h = V/πr2
⇒ h = 570/(3.14 × 4 × 4)
⇒ h = 11.34 m
Example 6: Calculate the height of a cylinder if its volume is 341 m3 and its radius is 6 m.
Solution:
We have,
V = 341 m3
r = 6 m
Using the formula we have,
V = πr2h
⇒ h = V/πr2
⇒ h = 341/(3.14 × 6 × 6)
⇒ h = 3.01 m
Practice Questions on Volume of Cylinder
Q1: Find Volume of Cylinder whose diameter is 14 cm and height is 12 cm.
Q2: Find Volume of Cylinder whose surface area of base is 84 cm2 and height is 11 cm.
Q3: Find the height of cylinder whose radius is 7 cm and volume is 770 cm3
Q4: Find the volume of a hollow cylinder of height 13 cm whose inner radius is 6 cm and outer radius is 1 cm.
Q5: A hollow cylindrical tube has an outer radius of 10 cm, an inner radius of 8 cm, and a height of 30 cm. Calculate the volume of the material used to make the tube.
Q6: Find the volume of an elliptic cylinder with a semi-major axis of 6 cm, a semi-minor axis of 4 cm, and a height of 10 cm.
Q7: The volume of an elliptic cylinder is 3,000 cm³, and the semi-major axis is 10 cm, while the height is 15 cm. Find the semi-minor axis.
Q8: Calculate the volume of an oblique cylinder whose base has a diameter of 14 cm and a perpendicular height of 12 cm.
Q9: The volume of an oblique cylinder is 5,000 m³, and the radius of its base is 20 m. What is the perpendicular height of the cylinder?
Q10: Find the volume of a hollow elliptic cylinder with an outer semi-major axis of 10 cm, an inner semi-major axis of 8 cm, and a height of 12 cm. The semi-minor axes are 6 cm and 4 cm, respectively.
|
Answers to Practice Questions
|
|
1. 1848.83 cm³
|
2. 923.02 cm³
|
3. 5 cm
|
4. 1429.97 cm³
|
5. 3393.05 cm³
|
|
6. 753.98 cm³
|
7. 6.37 cm
|
8. 1848.83 cm³
|
9. 3.98 m
|
10. 1055.58 cm³
|
Conclusion
Volume of a cylinder is essential for various real-world applications, from calculating the capacity of containers and tanks to solving problems in engineering and architecture. The formula V=πr2h provides a straightforward method to determine how much space a cylinder occupies, whether it’s a standard, hollow, or even an oblique or elliptic cylinder.
Learning this concept not only enhances your problem-solving skills but also offers practical benefits in everyday scenarios where accurate volume measurements are necessary.
Volume of Cylinder – FAQs
What is Volume of a Cylinder?
Volume of a cylinder is defined as the capacity of the cylinder, i.e. the amount of substance a cylinder can hold. It can also be defined as the total material required for making a cylinder.
Volume of the cylinder (V) is given by the formula, V = (Area of Circular Base) × (Height) Or V = πr2h
What is Volume of a Cylinder if its Radius is Doubled?
As the volume of the circle is proportional to the square of the radius. So if the radius of the circle is doubled then its volume becomes four times.
What is Volume of Cylinder if its Radius is halved?
As the volume of the circle is proportional to the square of the radius. So if the radius of the circle is halved then its volume becomes one by four times.
What is unit of Volume of a Cylinder?
Volume of a cylinder is measured in cubic units, i.e. cubic centimeters (cm3), cubic meters (m3), cubic feet (ft3) and so on for mathematical purposes. In general usage, it is also measured in Litters (l), milliliters (ml), etc.
What is Volume of a Hollow Cylinder?
If R is the external radius and r is the internal radius, then the formula for calculating the cylinder’s volume is given by: V = π (R2 – r2) h cubic units.
What is Volume of an Oblique Cylinder?
Formula to find the volume of an oblique cylinder is same as the volume of the cylinder and the volume of the oblique cylinder is calculated using the formula. V = π r2 h cubic units.
What is Volume of an Elliptical Cylinder?
Formula to find the volume of an Elliptical Cylinder is, V = πabh cubic units. Where, a and b are the radii of bases of the cylinder and h is the height of the cylinder.
What is Volume of a Solid Cylinder?
Volume of Solid Cylinder is the volume of material used in making the cylinder. For any solid cylinder of height ‘h’ and radius ‘r’ its volume is given by the formula, V = πr2h.