# Volume of a Cylinder

**Volume of a cylinder** is defined as the space or region which is enclosed by the cylinder or we can consider it as the amount of material the cylinder holds. The volume of the cylinder can be calculated by using the formula **πr ^{2}h** where r is the radius of the base and h is the height of the cylinder. The formula for finding the volume of a cylinder is explained in detail in this article.

## What is the Volume of a Cylinder?

Volume of a cylinder is the total capacity of any cylinder, i.e. it is the total amount of liquid any cylinder can hold. It is generally measured in liters. it can also be measured in m^{3}, cm^{3}, etc. It is calculated by multiplying the area of the base of the cylinder by its height.

### Cylinder Definition

A cylinder is a three-dimensional solid figure with two identical circular bases connected by a curving surface. It can be visualized as a stack of circular discs which are placed on top of each other. The perpendicular distance between the bases is called the height of a cylinder. The radius of the cylinder is actually the radius of any of its bases. The volume of the cylinder is nothing but the amount of substance this 3-D figure can hold. Thus, the volume of any 3-D figure is actually the space or material occupied by its surfaces.

## Volume of a Cylinder Formula

The volume of a cylinder is 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. The number of unit cubes (cubes of unit length) that may fit into a cylinder is its volume. Let us take a cylinder with height **h** and base radius of **r** then its volume is given by

V = πr^{2}hWhere,

ris the radius of base,his the height of cylinder.

**Derivation of Volume Formula for Cylinder**

The volume of any shape is equal to the product of its base area and height. Suppose we have a cylinder of radius r and height h. We know that the cylinder has a shape of its base similar to a circle.

So, the area of its circular base (A) = πr^{2}

Volume of the cylinder (V) is given by,

V = (Area of circular base) × (Height)

V = A×h

V = πr^{2}hwhere,

ris radius of base

his height of cylinder

**Volume of a Hollow Cylinder**

A hollow cylinder is a cylinder that is hollow from the inside thus, its bases form a ring with two radii inner radius and an outer radius. Suppose a hollow cylinder is taken with its inner radius as r_{1 }and outer radius as r_{2 }and the height of the cylinder is h then its volume is given by

V = πh(r_{1}^{2}– r_{2}^{2})Where,

ris the inner radius of base,_{1}ris the outer radius of base_{2}his the height of cylinder.

**Volume of a Cylinder in Liters**

Generally, the volume of a cylinder is calculated in cubic meters or cubic centimeters but we can change them in liters by using the conversion discussed below i.e.,

1 cm

^{3 }= 1 ml,1000 cm

^{3 }= 1 litre,1 m

^{3}= 1000000 cm^{3}= 1000 litres

**Example**: If a cylinder has a volume of 32 m^{3} it can be written as 32×1000 liters = 32000 liters

## Volume of a Right Circular Cylinder

For a right circular cylinder, the base is a circle with radius r and area πr^{2}. Then, the volume (V) of a right circular cylinder is given by

Volume(V) = Area of base × Height

V = πr^{2}hHere,

‘r’is the radius of the base of the cylinder‘h’is the height of the cylinder

**How To Find the Volume of a Cylinder?**

Let’s take an example to understand how we can calculate the volume of a cylinder.

**Example:** Calculate the volume of a cylinder of radius 4 m and height 5 m.

Step 1:Note the radius and height of cylinder. In this example, the radius of the cylinder is 4 m and height is 5 m.

Step 2:We know that the volume of a cylinder is equal to πr^{2}h. Substitute the given value of radius and height in the formula.

Step 3:So, the volume of cylinder is calculated as, 3.14 × (4)^{2}× 5 = 251.20 cu. m.

## Surface Area of a Cylinder

Surface Area of any figure is defined as the area required to totally cover the surface of any object. So, the surface area of a cylinder can be defined as the area required to cover the surface of the cylinder. Let the base radius of the cylinder is **r** and the height of the cylinder be **h** then its area is given by,

Curved Surface Area(CSA) = 2πrh

Total Surface Area(TSA) = 2πr(r+h)

## Solved Problems on Volume of a Cylinder

**Problem 1: Calculate the volume of a cylinder of radius 3 m and height of 4 m. (take π = 3.14)**

**Solution:**

We have, r = 3 and h = 4

Using the formula we have,

V = πr

^{2}h= 3.14 × (3)

^{2}× 4= 113.04 cu. m

**Problem 2: Calculate the volume of a cylinder of radius 4 m and height of 7 m.**

**Solution:**

We have, r = 4 and h = 7

Using the formula we have,

V = πr

^{2}h= 3.14 × (4)

^{2}× 7= 351.68 cu. m

**Problem 3: Calculate the radius of a cylinder if its volume is 300 cu. m and its height is 7 m.**

**Solution:**

We have, V = 300 and h = 7

Using the formula we have,

V = πr

^{2}h=> r

^{2}= V/πh=> r

^{2}= 300/(3.14 × 7)=> r = 3.68 m

**Problem 4: Calculate the radius of a cylinder if its volume is 450 cu. m and height are 9 m.**

**Solution:**

We have, V = 450 and h = 9

Using the formula we have,

V = πr

^{2}h=> r

^{2}= V/πh=> r

^{2}= 450/(3.14 × 9)=> r = 12.52 m

**Problem 5: Calculate the height of a cylinder if its volume is 570 cu. m and radius are 4 m.**

**Solution:**

We have, V = 570 and r = 4

Using the formula we have,

V = πr

^{2}h=> h = V/πr

^{2}=> h = 570/(3.14 × 4 × 4)

=> h = 11.34 m

**Problem 6: Calculate the height of a cylinder if its volume is 341 cu. m and radius are 6 m.**

**Solution:**

We have,

V = 341

r = 6

Using the formula we have,

V = πr

^{2}h=> h = V/πr

^{2}=> h = 341/(3.14 × 6 × 6)

=> h = 3.01 m

## FAQs on Volume of Cylinder

**Question 1: What is meant by the volume of a cylinder?**

**Answer:**

The volume of a cylinder is defined as the capacity of the cylinder, i.e, the amount of substace a cylinder cand hold. It can also be defined as the total material required for making a cylinder.

Volume of the cylinder (V) is given by,

V = (Area of circular base) × (Height)

V = πr^{2}hwhere,

ris radius of base

his height of cylinder

**Question 2: By what factor is the volume of the cylinder changed if its radius is doubled?**

**Answer:**

We already know that volume of cylinder is directly proportional to the square of its radius.

(volume of cylinder ∝ Radius^{2})

So if the radius is doubled, (i.e., r = 2r), we get

V = πr^{2}h

= π(2r)^{2}h

= 4πr^{2}h.

Hence, the volume of cylinder becomes four times its initial value, when its radius is doubled.

**Question 3: By what factor is the volume of the cylinder changed if its radius is halved?**

**Answer:**

We already know that volume of cylinder is directly proportional to the square of its radius.

(volume of cylinder ∝ Radius^{2})

So if the radius is halved, (i.e., r = r/2), we get

V = π(r/2)^{2}h

= (πr^{2}h)/4

Thus, the volume of cylinder becomes one-fourth its initial value, when its radius is halved.

**Question 4: What is the unit for the volume of a cylinder?**

**Answer:**

Volume of a cylinder is normally measured in cubic units, i.e. cubic centimeters (cm

^{3}), cubic meters (m^{3}), cubic feet (ft^{3}) and so on for mathematical purposes. In general usage, it is measured in Liters(l), milliliters(ml), etc.

**Question 5: Find the volume of a cylinder with diameter ‘d’ units and height ‘h’ units.**

**Answer:**

If the diameter of a cylinder is d then its radius is given by

radius = (diameter)/2so r = d/2

V = πr^{2}h= π(d/2)^{2}h

V = πdh / 4 unit^{3}